Annegret Wagler On Critically Perfect Graphs - CiteSeerX
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Annegret Wagler
On Critically Perfect Graphs
Preprint SC 96-50 (December 1996)
ritically Perfect
raphs
Annegret Wagler December 1996
Absac A perfect graph is critical if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to wellknown classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness
Introduction We assume familiarity with basic notions of graph theory and consider finte, undirected graphs without loops or multipl edges; subgraphs are nodeinduced. To better understand a graph property V, it is often useful to investigate extremal cases, i.e., raphs having V but losing it by a small modification. We distinguish two types of such graphs: m i n i m a l graphs (they possess V but lose it by the deletion of any node) and c r i t i a l graphs (they have V but lose it by the deletion of any edge). The subject of our investigation is a rich and wellstudied graph property: p e r f e c t e s s B E R G E proposed to call a graph perfec if, fo each of its subgraphs G', th chromatic number x(G') equals the clique number U(G), otherwise the graph is i m p e r f e c t . Chordless cycles of ength at least four have been termed h o e s and thei c o p l e m e n t s a n t i h o l e s . Obviously, any graph t h a t contains an odd hole or an odd antihole is imperfect. B E R G E conjectured in [2] t h a a graph is perfect iff it contains neither odd holes nor odd antiholes as subgraphs, i e . , if th graph is B e r g e (Strong Perfect Graph Conjecture, for short S P G C ) . P A D B E R G [16] introduced th notion of imally imperfect graphs; in these terms, the S P G C states t h a t the odd holes and the odd antiholes are the only minimally imperfect graphs. Therefore, minimally imperfect Berge graphs are called m t e r s ( s c e the existence of this third type of imally imperfect raphs would contradict the SPGC)
A t h o u g h several, in general A/'P-hard, combinatorial optimization problems can be solved in polynomial time fo perfect graphs, see [8], the structure of perfect graphs is not well understood. In particular, th S P G C still seems to be out of reach. On the other hand, th investigation of imally imperfect graphs has revealed t h a t these graphs have quite strong properties, see e.g. [5, 9, 13, 14, 15, 16, 18, 19]. T h a t motivated us to introduce a new class of e x r e m a l cases with respect to p e r f e c t e s s : critically perfect graphs. W provide several examples and prove some basi properties n section 2. We are interested in relating the class CV of critically perfect graphs to wellknown classes of perfect graphs. It turns out t h a t many linegraphs of i p a r t t e graphs as well as complements
of s c h graphs belong o CV; let us denote this subset of CV by LQ. ince there is no cla of perfect graphs known which contains both the linegraphs of bipartite graphs and thei c o p l e m e n t s , it seems hopeless to find a class of perfect graphs c o n t a i n n g all graphs of CV. In order to examin the structure of the class CV, we first characterize the graphs belonging to LQ (see section 3). Then we study operations preservng critical p e r f e c t e s s in section 4. We btain a subset of CV which can be constructed fro graphs in LQ by applying this kind of perations. Finally, w check when c o p l e m e n t a t i o preserves critical p e r f e c t e s s
ritical Perfectness We define an edge e 6 E of a perfect graph G = (V, E) to be critical if G — e is imperfect. In p a r t c u l a r , for every critical edge e of a perfect graph G, there is a subgraph Ge C G s t . Ge — e is imally mperfect According to th three different types of minimally imperfect graphs, we distinguish between three types of critical edges. We say t h a t an edge e of a perfect graph G is H-critical (Acritical, M - c r i t i a l , resp. if G — e contains an odd hole (an odd antihole, a monster, resp.) Note t h a t an H-critical edge is a single chord in an odd cycle of length > 5 which forms a riangl with two edges of this cycle; further, if the S P G C is true, there are no M-critical edges We define a graph to be c r i t i a l l y perfec if is a perfect graph without isolated nodes and all of ts edges are critical Critical perfectness is a very strong property. Nevertheless, there are surprisingly many graphs having this property; a few examples are shown in Figure note t h a t Gi GQ are linegraphs of i p a r t e graphs"!" a n c [ Q1 = Q1 = Q4 = Q4 holds) Some p r o p e r t e s of minimally imperfect graphs immediately determin properties of a critically perfect graph G: if x and y are adjacent nodes of G, then neither x d o m i n a t e s y, i.e., N(y) C N(x) U {x} (easy) nor do all induced paths connecting x and y in G — xy have even length ( M E Y N I E L [14]), nor are all nodes in G — {x,y} adjacent either to x or to y ( O L A R I U [15]). I addition, we state the following basic properties L e m m a 2.1 For any critically perfect graph G, it holds that (i) every edge of G is contained in a triangle (ii) there is no simplicial (ii) G has minimal
node in G, ie,
a node x st.
degree 5(G) > 4 and maximal
N(x)
is a clique
degree A(G) < n — 3.
Proof. Let G = (V, E) be a critically perfect graph. (i) Suppose e 6 E is not contained in a riangle. Then e is either H - c r i c a l nor critical Hence, there is G e C G s t . G e — e is a monster. LovÄsz's characterization of perfect graphs [13] says a(Ge)uj(Ge) > n but a(Ge — e)uj(Ge — e) < n. Therefore, u(Ge — e) < u(Ge) (not t h a t üj(Ge) > (Ge — e) and a(Ge) < a(G — e)) holds and e is contained in the intersection of all maximum cliques of Ge. It follows O(G) = 2, since e is a maximal clique of G, a contradictio to LL> > 3 for any monster by T U C K E R [19]. O
*A well-known characterization of linegraphs of b i p a r t i t e graphs is t h a t one can colour their edges with olours s t . e v e y m a x i m a l clique is m o n o c h r o m a t i c a n d no two m a x i m a l cliques of t h e same colour i n t t
G1
G2
G3
G1
G2
G3
G4
G5
G6
G4
Gg
G^
Figure 1: Examples fo critically perfect graphs
(ii) Since G is noncomplete, a node y in t e cliqu N(x) would have a eighbour in G — (N(x) U {x}), i e . , y would dominate x, a contradiction.O (iii) We first show 5(G) > 4. Edges ncident to nodes with degree < 3 are not critcal Assum t h a t there is x G V with N(x) = {21,22,2:3}. Since 5(C2k+i) > 4 for k > 2 and 5(M) > 6 fo any monster M holds by S E B Ö [18] and [19], the edges incident to x are H-critical. Consider th edge 2:2:1, it has to be contained in a riangle. Without loss of generality, let 2:122 be an edge. Further, 2:1, x, 2:3 are contained in an even hole C with x^ ^ G and Nc(x2) = {x,x}, i.e., 2:123,222:3 G E. But now, 2:23 is not contained in a triangle, a contradiction to (i). Now, we show A(G) < n — 3. A node of degree n — 1 would dominate all other nodes. Consider G V with d() = n — 2 and - G- N() ince tit G- -E, u G . / V ) would imply t h a t dominates v, we have A(tt) = N(w) et e be an edge i c i d e n t to and Ge — e the coresponding minimally imperfect subgraph. By M E Y N I E L [14], it follows w G- Ge — e, i e . , Ge C V(M) U { and, in Ge — e, is contained i exactly o e stable set of size 2, a contradictio to Ge — minimally imperfect. • Note t h a t the bounds for 5(G) and A(G) in (iii) are sharp (this is show by the graph G Figure 1 and by the c o p l e m e n t of the linegraph of the second graph in Figure 3, resp.)
Characterization of the
in
aphs in LQ
n order to characterize the graphs in CQ, we describe two structures in the underlying bipart raphs F which guarantee t h a t the edges of L(F) and L(F), respectively, are critical We say t h a t two incident edges x and y form an H-pai if there is a K\^ with edges x, y, z and an even cycle C t h a t contains x and y but only o e endnode of z. We call F an H - g r a p h if i is connected and every two of its i c i d e n t edges form an H-pair Examples of H-graphs are th ipartite graphs shown in Figure 2 and all 3-connected b i p a r t e graphs easy)
Figure 2: Examples for H-graphs We say t h a t two n o i n c i d e n t edges form an A-pair if they are th end edges of an odd path with length at least five. We call F an A - g r a p h if it is connected and every two of its n o i n c i d e n t edges form an A-pair Examples for A g r a p h s are th partite graphs shown in Figure 3 and all 3-connected b i p a r t e graphs (easy)
Vy Figure 3: Examples fo
graphs
Let us first investigate t e r e l a t i o n s i p of t e d e f i n d s t c t u r e in H - a p h s wi in linegraphs
cri
dge
T h e o r e m 3.1 Let G be a perfect graph and the Unegraph of a graph F. An edge e = xy of G is critical iff x and y form an H-pair in F. P r o o f (If) Let x and y form an H-pair in F, then there is an edge z incident to both and an even cycl C c o n t a n i n g x and y but only one endnode of z. In G = L(F), the edge e = xy is contained in the even hole L(C) and it holds N^tc\(z) = {x,y}, i e . , the edge e is H-critical. O (Only if) Let G be perfect, the linegraph of a graph F, and e a critical edge of G. We show t h a t th edges corresponding to the endnodes of e form an H-pair in F. ccording to the three types of critical edges, we distinguish between three cases Case 1: e is H-critical Let e = xy, the nodes x and y are contained n an even hol Gik, k > 2 and there is another node z with Nc2k(z) = {x, y}. In F, x and y are incident edges on the even cycl L~(C2k)] is incident to x and y and has exactly one endpoint on L~(C2k) (else z has four neighbours on C)Thus, x and y form an H - p a r in F. Case 2: e is critical et e = v\V2k+i and v\..., t2fc+i be th nodes of the corresponding odd antihole C'2k+i,k > in G — e with t w + ^ E(G) fo < i < 2&. For k > 3, th nodes , t>2j u 3 5 u2fc5 w fc+i induce a subgraph of C which is forbidden according to B E I N E K E ' S characterization of linegraphs [1]. Hence, it holds k = 2 and the assertion follows by Case 1, since e is also H-critical not C5 + e ~ C5 + e) Case 5: e is M-critical Let e = xy and M — e be a monster. By [1], M is Ä'i j 3 free but M — e is not since th S P G C is true for Ä ' i f r e e graphs by a result of P A R T H A S A R A T H Y and R A V I N D R A [17]. et us now consider a Ä'i j3 = {w, x, y; z} with center z in — e. Th nodes x, z, w and y,z,w, resp., induce i V s which are contained in the holes Cx and Cy, resp. (see HoÄNG [9]) Since M — e is Berge, Cx and Cy are even holes. L~l(Cx) and L _ 1 (Cj / ) are even cycles in F and C = (Cx U C y ) — {w, z} is an even cycle containing the edges x, y and only one endnode of z. But now, L(C U {z}) — e is an odd hole in M — e, a contradictio to M — e Berge. • We o t a i n the following corollary as an immediate consequence of the previous proof C o r o l l a y 3.2 Let G be perfect and a linegraph. An edge of G is critical iff it is h e o r e m 3.3 Let G be the linegraph of F. G is critically perfect i
H-critical
F is a bipartite
H-graph
Proof (If) Let F be a bipartite H-graph, then its linegraph C is perfect by KÖNlG' Edge-Colouring-Theorem 11]; every edge of C is c r i c a l by Theorem 3.1. O (Only if) et C be criically perfect and the linegraph of F. We know by Theorem 3.1 t h a t th edges of F corresponding to adjacent nodes in C form an H-pair. Thus F is an H-graph. W have to show t h a t F is b i p a r t t e . Assume F contains an odd cycle. Since C contains an odd hole if this cycle has a length > 5, it has to be a triangle with edges e\, e2, e^. Since F is an H-graph, e\ and C2 appear on an even cycle C t h a t has to be a C (else (C — {e\, C2}) U {e 3 } would be an odd cycle of length > 5) et ei3 be th edge of C incident to e\ and e 3 , and e2 3 be the edge of C incident to C and e 3 . Bu e\ and e also appear on an even cycl C and we get an odd cycle of length > 5 in each case: (C" - {e 3 }) U { e i , e } if e 3 ^ C, or ( - {e 2 3 , e 3 }) U {ei 3 } if e G C". Since the existence of an odd cycle in F contradicts the p e r f e c t e s s of C, F has to be a ipartite graph. •
Now, let us relate t e s c t u r e in similar way.
aphs t
crial
dges i
plements of l i n a p h s in a
T h e o r e m 3.4 Let G be perfect and the complement of the Unegraph of a graph F. e = xy of G is critical iff' x and y form an A-pair in F.
An edge
Proof. If) et x and y form an A-pair in F, then they are the end edges of an path P2k, k > 3. It holds P2) = Ci-\ + e where e is an additional edge between these nodes of i whch e correspond to the end edges of th path P2k-, i - 5 the edge e is critical. O (Only if) Let G be perfect the complement of the linegraph of F, and e a critical edge of G. We show t h a t the edges corresponding to the endnodes of form an A-pair in F. ccording to the three types of critical edges, we distinguish between three cases Case 1: e is -critical. Let e v\Vk+\ and v\ ..,V2k+i be the nodes of the corresponding odd antihol C2k+i,k > 2 in G — e with w + i ^ E(G) for 1 < i < 2k. It holds w + G E(G) fo 1 < i < 2&, i e . , i>i and t2fc+i a r e the endnodes of a path fc+i = C2fc+ + e in G, hence th end edges of path with length 2k + 1 in F . Case 5: e is H-critical. Let e = v\V2k and u , . . . i2fc+i be the nodes of th corresponding odd hole C2k+ik > 2 in G - e with vtv,+1 G E(G) for 1 < i < 2k + l(mod2A; + 1 ) . Fo & > 3, the nodes v\, v3, V2k, ^2fc+i induce a Ü'i^ in G which is forbidden by [1] (note t h a G is Ä ' i f r e e ) Hence, it holds k = 2 and the assertion follows by Cas 1, since e is also critical Case 3: e is M-critical et e = xy and M — e be a monster. By [1], M is Ä'i^-free but M — e is not since M + e is a monster as well by LovÄsz's Perfect Graph Theorem [13], and the S P G C is true for Ä'i^free graphs by [17] Let us now consider a K\^ {w, x, z; y} in M — e, i e . , a Ü'1,3 with center y in M + e. The nodes w, y, z and x,y,z, resp. nduce P 3 ' i n M + e w h c h are contained in the holes Cw and Cx, resp. (see [9]). Since M + e is Berge, Cw and are even holes In M, we have C w and an even path Px connecting z and x. Finally, P = L(C U Px) — {z} is a path of odd length > 5 with x and y as end edges; but L(P) is an odd antihol n M — e, a contradictio to M — e Berge. • Corollary 3.5 _e£ G 6e perfect and the complement it is A-critical T h e o r e m 3.6 Let G be perfect and the complement perfect i F is a bipartite A-graph Proof. KONIG'
of a linegraph. An edge of G is critical i
of the linegraph of a graph F. G is critically
If) Let F be a bipartite A-graph, then th c o p l e m e n t G of its l e g r a p h is perfect by Matching-Theorem [12]; every edge of G is critical by Theorem 3.1.
(Only if) Let G be critically perfect and the complement of the linegraph of F. We know by Theorem 3.4 t h a t the edges of F corresponding to adjacent nodes in G form an A-pair. Thus F is an A-graph. We have to show t h a t F is bipartite. ssume F contains an odd cycle D. Since G contains an odd antihole if D has a length > 5, D has to be a riangle with nodes x,y,z. Since F contains nonincident edges (note t h a t G is not a stable set) and is edgeconnected, there is an edge incident to D, say x' is adjacent to x. The edges xx' and yz form an A-pair, i e . , they are the end edges of an odd path P with length > 5 and we get an odd cycle G with length > 5: G = {xx'y} if P = {xx'yz}, or G = { x . } if P = {x'x.z}, a contradiction. •
erations
reserving
ritical Perfectness
this section, we study graph operations with th property t h a t the class CV is closed under applying these operations graph operation transforms graphs G\, .,Gi into a new graph G. If an operation ransfers a c o m o n property V of G\, ,Gi to G, we say t h a t this operation preserves th property V. et us call a perfectio preserving operation a P - o p e r a t i o n . We define a P * - o p e r a t i to be P-operatio t h a t generates G fro disjoint graphs G\ and G s t (i) G i C G and G C G (ii) e G E{d) t follows fro Lemma
or e G £ ( G ) Ve G £ ( G ) the d e f i n i o
t h a t the class CV is closed under a p p l y n g P * o p e r a t i o n s
.1 Every P*operation
preserves
critical
perfectness
et us give some examples of P*-operations. Obviously, the union and all perfection preserving identification operations are P * o p e r a t i o n s : the cliqu identification; ts generalization, th subgraph identificatio by Hsu [10]; and the stable set identification by a result of C O R N E I L and F O N L U P T [6]. Furthermore, the multiplication and ts generalization, the substitution by LovÄsz's Replacement emma [13], and th c o p o s i t i o by BlXB [3] are P * o p e r a t i o n s as ell Let us mention that, if G L(F), th multiplcation of a node v G V(G) is similar to th addition of parallel edges to v in the underlying graph F. Further, since no node of a critically perfect graph dominates o e of its neighbours, no critically perfect graph satisfies th conditio of the amalgam operation introduced by B U R L E T and F O N L U P T [4], exept for the special case of B I X B Y ' S composition. The same holds for all further generalizations of the amalgam, e.g. the so-called 2-amalgam defined by C O R N U E J O L S and C U N N I N G H A M [7]
Finally, note t h a t a graph G generated from two graphs G i , G 2 G CQ by P * o p e r a t i o n s does not belong to LQ in general, as is shown by the two examples in Figure 4.
Figure 4: E x a m p l s for graphs genera
by P * o p e r a i o n s
et us now check, when complementaion, which preserves p e r f e c t e s s by LovÄsz's Perfect Graph Theorem [13], preserves critcal perfectness. Both a graph and ts complement are crit cally perfect if the deletion and the a d d i i o n of any edge results in an imperfect graph. This requirement is very strong. Indeed, the elements of CV have noncriical c o p l e m e n t s in general i e . , CV is not closed under c o p l e m e n t a t i o n . E.g., all linegraphs of ipartite graphs t h a t are not also H-graphs (see the linegraphs of th A g r a p h s shown in Figure 3), are n o c r i t i c a l
Obviously, he same holds for all graphs generate with n o c r i t i c a l complement
by P * o p e r a t i o n s fro
t l s t o e graph
However, there are many examples of c r i c a l l y perfect graphs with critical complements, e.g. the graphs show n Figure 1 and in Figure 4. We are able to prove some necessary conditions t h a t th c o p l e m e n t of a critically perfect graph is critical as well Lemma
2 If the complement
(i) is a connected
of a critically perfect graph G is critical, then G
graph,
(ii) has no cliquecutset v G E{G Vv G (ii) has no nonadjacent
t in at least two components
nodes with the same
ofG —
there exists a node x with
neighbourhood.
Proof. Let G be a critically perfect graph. (i) Assume t h a t G is disconnected with components G\ and G^ G, consider an edge e = x\X2 with x\ G V(G\), x G ViGq) and let Ge — e be the corresponding minimally imperfect subgraph of G — e. By LovÄsz [13], Ge + e C G + e is m i m a l l y imperfect as well, and e is the only edge connecting (Ge + e) n G\ and (Ge + e) n G- Therefore, Ge + e cannot be (2UJ — 2)connected, a contradiction to S E B Ö [18]. O (ii) Assume t h a Q is a cliquecutset of G, G\ and i are two c o p o n e n t s of G — Q, and x\ G G i , X2 G G are nodes with x\v, x^v G E(G) \/v G Q. In G, consider the edge e = X\X2 and the minimally imperfect subgraph Ge — e. By [13], Ge + e C G + e is minimally imperfect. Bu G e + e fl (Q U {x\ X) is a cliquecutset of Ge + e, a contradictio to C H V A T A L ' S Star-Cutset Lemma [5]. O (ii) Assum t h a t x\ and X2 are nonadjacent nodes with the same neighbourhood; in G, they are adjacent nodes and would dominate each other, a contradiction. • Therefore, either the c o p l e m e n t s of graphs generated by the unio (see emma 4.2(i)) no by the identificatio in a node or an edge (see Lemma 4.2(ii)) nor by the multiplicatio (see Lemma 4.2(ii)) are critically perfect. Furthermore, it follows by emma 4.2(i) t h a t th ellknown c o p l e t e join does not preserve critical p e r f e c t e s s We are further interested in - sufficient conditions t h a t guarantee the critical p e r f e c t e s s of a raph and ts c o p l e m e n t critically perfect graphs t h a t cannot be c o n s r u c t e d fro P*operations o coplementation,
the graphs in CQ by applying
- additional examples for P * o p e r a t i o n s , and - further operations preserving o c o n s r u c t i n g critical p e r f e c t e s s
Ackowedgent The autho is grateful to Prof. M. Grötschel for his suggestio perfect graphs and for his valuabl remarks
to investigate th class of critically
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Graphs. J. C o i n . Theory
Annegret Wagler
On Critically Perfect Graphs
Preprint SC 96-50 (December 1996)
ritically Perfect
raphs
Annegret Wagler December 1996
Absac A perfect graph is critical if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We relate critically perfect graphs to wellknown classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness
Introduction We assume familiarity with basic notions of graph theory and consider finte, undirected graphs without loops or multipl edges; subgraphs are nodeinduced. To better understand a graph property V, it is often useful to investigate extremal cases, i.e., raphs having V but losing it by a small modification. We distinguish two types of such graphs: m i n i m a l graphs (they possess V but lose it by the deletion of any node) and c r i t i a l graphs (they have V but lose it by the deletion of any edge). The subject of our investigation is a rich and wellstudied graph property: p e r f e c t e s s B E R G E proposed to call a graph perfec if, fo each of its subgraphs G', th chromatic number x(G') equals the clique number U(G), otherwise the graph is i m p e r f e c t . Chordless cycles of ength at least four have been termed h o e s and thei c o p l e m e n t s a n t i h o l e s . Obviously, any graph t h a t contains an odd hole or an odd antihole is imperfect. B E R G E conjectured in [2] t h a a graph is perfect iff it contains neither odd holes nor odd antiholes as subgraphs, i e . , if th graph is B e r g e (Strong Perfect Graph Conjecture, for short S P G C ) . P A D B E R G [16] introduced th notion of imally imperfect graphs; in these terms, the S P G C states t h a t the odd holes and the odd antiholes are the only minimally imperfect graphs. Therefore, minimally imperfect Berge graphs are called m t e r s ( s c e the existence of this third type of imally imperfect raphs would contradict the SPGC)
A t h o u g h several, in general A/'P-hard, combinatorial optimization problems can be solved in polynomial time fo perfect graphs, see [8], the structure of perfect graphs is not well understood. In particular, th S P G C still seems to be out of reach. On the other hand, th investigation of imally imperfect graphs has revealed t h a t these graphs have quite strong properties, see e.g. [5, 9, 13, 14, 15, 16, 18, 19]. T h a t motivated us to introduce a new class of e x r e m a l cases with respect to p e r f e c t e s s : critically perfect graphs. W provide several examples and prove some basi properties n section 2. We are interested in relating the class CV of critically perfect graphs to wellknown classes of perfect graphs. It turns out t h a t many linegraphs of i p a r t t e graphs as well as complements
of s c h graphs belong o CV; let us denote this subset of CV by LQ. ince there is no cla of perfect graphs known which contains both the linegraphs of bipartite graphs and thei c o p l e m e n t s , it seems hopeless to find a class of perfect graphs c o n t a i n n g all graphs of CV. In order to examin the structure of the class CV, we first characterize the graphs belonging to LQ (see section 3). Then we study operations preservng critical p e r f e c t e s s in section 4. We btain a subset of CV which can be constructed fro graphs in LQ by applying this kind of perations. Finally, w check when c o p l e m e n t a t i o preserves critical p e r f e c t e s s
ritical Perfectness We define an edge e 6 E of a perfect graph G = (V, E) to be critical if G — e is imperfect. In p a r t c u l a r , for every critical edge e of a perfect graph G, there is a subgraph Ge C G s t . Ge — e is imally mperfect According to th three different types of minimally imperfect graphs, we distinguish between three types of critical edges. We say t h a t an edge e of a perfect graph G is H-critical (Acritical, M - c r i t i a l , resp. if G — e contains an odd hole (an odd antihole, a monster, resp.) Note t h a t an H-critical edge is a single chord in an odd cycle of length > 5 which forms a riangl with two edges of this cycle; further, if the S P G C is true, there are no M-critical edges We define a graph to be c r i t i a l l y perfec if is a perfect graph without isolated nodes and all of ts edges are critical Critical perfectness is a very strong property. Nevertheless, there are surprisingly many graphs having this property; a few examples are shown in Figure note t h a t Gi GQ are linegraphs of i p a r t e graphs"!" a n c [ Q1 = Q1 = Q4 = Q4 holds) Some p r o p e r t e s of minimally imperfect graphs immediately determin properties of a critically perfect graph G: if x and y are adjacent nodes of G, then neither x d o m i n a t e s y, i.e., N(y) C N(x) U {x} (easy) nor do all induced paths connecting x and y in G — xy have even length ( M E Y N I E L [14]), nor are all nodes in G — {x,y} adjacent either to x or to y ( O L A R I U [15]). I addition, we state the following basic properties L e m m a 2.1 For any critically perfect graph G, it holds that (i) every edge of G is contained in a triangle (ii) there is no simplicial (ii) G has minimal
node in G, ie,
a node x st.
degree 5(G) > 4 and maximal
N(x)
is a clique
degree A(G) < n — 3.
Proof. Let G = (V, E) be a critically perfect graph. (i) Suppose e 6 E is not contained in a riangle. Then e is either H - c r i c a l nor critical Hence, there is G e C G s t . G e — e is a monster. LovÄsz's characterization of perfect graphs [13] says a(Ge)uj(Ge) > n but a(Ge — e)uj(Ge — e) < n. Therefore, u(Ge — e) < u(Ge) (not t h a t üj(Ge) > (Ge — e) and a(Ge) < a(G — e)) holds and e is contained in the intersection of all maximum cliques of Ge. It follows O(G) = 2, since e is a maximal clique of G, a contradictio to LL> > 3 for any monster by T U C K E R [19]. O
*A well-known characterization of linegraphs of b i p a r t i t e graphs is t h a t one can colour their edges with olours s t . e v e y m a x i m a l clique is m o n o c h r o m a t i c a n d no two m a x i m a l cliques of t h e same colour i n t t
G1
G2
G3
G1
G2
G3
G4
G5
G6
G4
Gg
G^
Figure 1: Examples fo critically perfect graphs
(ii) Since G is noncomplete, a node y in t e cliqu N(x) would have a eighbour in G — (N(x) U {x}), i e . , y would dominate x, a contradiction.O (iii) We first show 5(G) > 4. Edges ncident to nodes with degree < 3 are not critcal Assum t h a t there is x G V with N(x) = {21,22,2:3}. Since 5(C2k+i) > 4 for k > 2 and 5(M) > 6 fo any monster M holds by S E B Ö [18] and [19], the edges incident to x are H-critical. Consider th edge 2:2:1, it has to be contained in a riangle. Without loss of generality, let 2:122 be an edge. Further, 2:1, x, 2:3 are contained in an even hole C with x^ ^ G and Nc(x2) = {x,x}, i.e., 2:123,222:3 G E. But now, 2:23 is not contained in a triangle, a contradiction to (i). Now, we show A(G) < n — 3. A node of degree n — 1 would dominate all other nodes. Consider G V with d() = n — 2 and - G- N() ince tit G- -E, u G . / V ) would imply t h a t dominates v, we have A(tt) = N(w) et e be an edge i c i d e n t to and Ge — e the coresponding minimally imperfect subgraph. By M E Y N I E L [14], it follows w G- Ge — e, i e . , Ge C V(M) U { and, in Ge — e, is contained i exactly o e stable set of size 2, a contradictio to Ge — minimally imperfect. • Note t h a t the bounds for 5(G) and A(G) in (iii) are sharp (this is show by the graph G Figure 1 and by the c o p l e m e n t of the linegraph of the second graph in Figure 3, resp.)
Characterization of the
in
aphs in LQ
n order to characterize the graphs in CQ, we describe two structures in the underlying bipart raphs F which guarantee t h a t the edges of L(F) and L(F), respectively, are critical We say t h a t two incident edges x and y form an H-pai if there is a K\^ with edges x, y, z and an even cycle C t h a t contains x and y but only o e endnode of z. We call F an H - g r a p h if i is connected and every two of its i c i d e n t edges form an H-pair Examples of H-graphs are th ipartite graphs shown in Figure 2 and all 3-connected b i p a r t e graphs easy)
Figure 2: Examples for H-graphs We say t h a t two n o i n c i d e n t edges form an A-pair if they are th end edges of an odd path with length at least five. We call F an A - g r a p h if it is connected and every two of its n o i n c i d e n t edges form an A-pair Examples for A g r a p h s are th partite graphs shown in Figure 3 and all 3-connected b i p a r t e graphs (easy)
Vy Figure 3: Examples fo
graphs
Let us first investigate t e r e l a t i o n s i p of t e d e f i n d s t c t u r e in H - a p h s wi in linegraphs
cri
dge
T h e o r e m 3.1 Let G be a perfect graph and the Unegraph of a graph F. An edge e = xy of G is critical iff x and y form an H-pair in F. P r o o f (If) Let x and y form an H-pair in F, then there is an edge z incident to both and an even cycl C c o n t a n i n g x and y but only one endnode of z. In G = L(F), the edge e = xy is contained in the even hole L(C) and it holds N^tc\(z) = {x,y}, i e . , the edge e is H-critical. O (Only if) Let G be perfect, the linegraph of a graph F, and e a critical edge of G. We show t h a t th edges corresponding to the endnodes of e form an H-pair in F. ccording to the three types of critical edges, we distinguish between three cases Case 1: e is H-critical Let e = xy, the nodes x and y are contained n an even hol Gik, k > 2 and there is another node z with Nc2k(z) = {x, y}. In F, x and y are incident edges on the even cycl L~(C2k)] is incident to x and y and has exactly one endpoint on L~(C2k) (else z has four neighbours on C)Thus, x and y form an H - p a r in F. Case 2: e is critical et e = v\V2k+i and v\..., t2fc+i be th nodes of the corresponding odd antihole C'2k+i,k > in G — e with t w + ^ E(G) fo < i < 2&. For k > 3, th nodes , t>2j u 3 5 u2fc5 w fc+i induce a subgraph of C which is forbidden according to B E I N E K E ' S characterization of linegraphs [1]. Hence, it holds k = 2 and the assertion follows by Case 1, since e is also H-critical not C5 + e ~ C5 + e) Case 5: e is M-critical Let e = xy and M — e be a monster. By [1], M is Ä'i j 3 free but M — e is not since th S P G C is true for Ä ' i f r e e graphs by a result of P A R T H A S A R A T H Y and R A V I N D R A [17]. et us now consider a Ä'i j3 = {w, x, y; z} with center z in — e. Th nodes x, z, w and y,z,w, resp., induce i V s which are contained in the holes Cx and Cy, resp. (see HoÄNG [9]) Since M — e is Berge, Cx and Cy are even holes. L~l(Cx) and L _ 1 (Cj / ) are even cycles in F and C = (Cx U C y ) — {w, z} is an even cycle containing the edges x, y and only one endnode of z. But now, L(C U {z}) — e is an odd hole in M — e, a contradictio to M — e Berge. • We o t a i n the following corollary as an immediate consequence of the previous proof C o r o l l a y 3.2 Let G be perfect and a linegraph. An edge of G is critical iff it is h e o r e m 3.3 Let G be the linegraph of F. G is critically perfect i
H-critical
F is a bipartite
H-graph
Proof (If) Let F be a bipartite H-graph, then its linegraph C is perfect by KÖNlG' Edge-Colouring-Theorem 11]; every edge of C is c r i c a l by Theorem 3.1. O (Only if) et C be criically perfect and the linegraph of F. We know by Theorem 3.1 t h a t th edges of F corresponding to adjacent nodes in C form an H-pair. Thus F is an H-graph. W have to show t h a t F is b i p a r t t e . Assume F contains an odd cycle. Since C contains an odd hole if this cycle has a length > 5, it has to be a triangle with edges e\, e2, e^. Since F is an H-graph, e\ and C2 appear on an even cycle C t h a t has to be a C (else (C — {e\, C2}) U {e 3 } would be an odd cycle of length > 5) et ei3 be th edge of C incident to e\ and e 3 , and e2 3 be the edge of C incident to C and e 3 . Bu e\ and e also appear on an even cycl C and we get an odd cycle of length > 5 in each case: (C" - {e 3 }) U { e i , e } if e 3 ^ C, or ( - {e 2 3 , e 3 }) U {ei 3 } if e G C". Since the existence of an odd cycle in F contradicts the p e r f e c t e s s of C, F has to be a ipartite graph. •
Now, let us relate t e s c t u r e in similar way.
aphs t
crial
dges i
plements of l i n a p h s in a
T h e o r e m 3.4 Let G be perfect and the complement of the Unegraph of a graph F. e = xy of G is critical iff' x and y form an A-pair in F.
An edge
Proof. If) et x and y form an A-pair in F, then they are the end edges of an path P2k, k > 3. It holds P2) = Ci-\ + e where e is an additional edge between these nodes of i whch e correspond to the end edges of th path P2k-, i - 5 the edge e is critical. O (Only if) Let G be perfect the complement of the linegraph of F, and e a critical edge of G. We show t h a t the edges corresponding to the endnodes of form an A-pair in F. ccording to the three types of critical edges, we distinguish between three cases Case 1: e is -critical. Let e v\Vk+\ and v\ ..,V2k+i be the nodes of the corresponding odd antihol C2k+i,k > 2 in G — e with w + i ^ E(G) for 1 < i < 2k. It holds w + G E(G) fo 1 < i < 2&, i e . , i>i and t2fc+i a r e the endnodes of a path fc+i = C2fc+ + e in G, hence th end edges of path with length 2k + 1 in F . Case 5: e is H-critical. Let e = v\V2k and u , . . . i2fc+i be the nodes of th corresponding odd hole C2k+ik > 2 in G - e with vtv,+1 G E(G) for 1 < i < 2k + l(mod2A; + 1 ) . Fo & > 3, the nodes v\, v3, V2k, ^2fc+i induce a Ü'i^ in G which is forbidden by [1] (note t h a G is Ä ' i f r e e ) Hence, it holds k = 2 and the assertion follows by Cas 1, since e is also critical Case 3: e is M-critical et e = xy and M — e be a monster. By [1], M is Ä'i^-free but M — e is not since M + e is a monster as well by LovÄsz's Perfect Graph Theorem [13], and the S P G C is true for Ä'i^free graphs by [17] Let us now consider a K\^ {w, x, z; y} in M — e, i e . , a Ü'1,3 with center y in M + e. The nodes w, y, z and x,y,z, resp. nduce P 3 ' i n M + e w h c h are contained in the holes Cw and Cx, resp. (see [9]). Since M + e is Berge, Cw and are even holes In M, we have C w and an even path Px connecting z and x. Finally, P = L(C U Px) — {z} is a path of odd length > 5 with x and y as end edges; but L(P) is an odd antihol n M — e, a contradictio to M — e Berge. • Corollary 3.5 _e£ G 6e perfect and the complement it is A-critical T h e o r e m 3.6 Let G be perfect and the complement perfect i F is a bipartite A-graph Proof. KONIG'
of a linegraph. An edge of G is critical i
of the linegraph of a graph F. G is critically
If) Let F be a bipartite A-graph, then th c o p l e m e n t G of its l e g r a p h is perfect by Matching-Theorem [12]; every edge of G is critical by Theorem 3.1.
(Only if) Let G be critically perfect and the complement of the linegraph of F. We know by Theorem 3.4 t h a t the edges of F corresponding to adjacent nodes in G form an A-pair. Thus F is an A-graph. We have to show t h a t F is bipartite. ssume F contains an odd cycle D. Since G contains an odd antihole if D has a length > 5, D has to be a riangle with nodes x,y,z. Since F contains nonincident edges (note t h a t G is not a stable set) and is edgeconnected, there is an edge incident to D, say x' is adjacent to x. The edges xx' and yz form an A-pair, i e . , they are the end edges of an odd path P with length > 5 and we get an odd cycle G with length > 5: G = {xx'y} if P = {xx'yz}, or G = { x . } if P = {x'x.z}, a contradiction. •
erations
reserving
ritical Perfectness
this section, we study graph operations with th property t h a t the class CV is closed under applying these operations graph operation transforms graphs G\, .,Gi into a new graph G. If an operation ransfers a c o m o n property V of G\, ,Gi to G, we say t h a t this operation preserves th property V. et us call a perfectio preserving operation a P - o p e r a t i o n . We define a P * - o p e r a t i to be P-operatio t h a t generates G fro disjoint graphs G\ and G s t (i) G i C G and G C G (ii) e G E{d) t follows fro Lemma
or e G £ ( G ) Ve G £ ( G ) the d e f i n i o
t h a t the class CV is closed under a p p l y n g P * o p e r a t i o n s
.1 Every P*operation
preserves
critical
perfectness
et us give some examples of P*-operations. Obviously, the union and all perfection preserving identification operations are P * o p e r a t i o n s : the cliqu identification; ts generalization, th subgraph identificatio by Hsu [10]; and the stable set identification by a result of C O R N E I L and F O N L U P T [6]. Furthermore, the multiplication and ts generalization, the substitution by LovÄsz's Replacement emma [13], and th c o p o s i t i o by BlXB [3] are P * o p e r a t i o n s as ell Let us mention that, if G L(F), th multiplcation of a node v G V(G) is similar to th addition of parallel edges to v in the underlying graph F. Further, since no node of a critically perfect graph dominates o e of its neighbours, no critically perfect graph satisfies th conditio of the amalgam operation introduced by B U R L E T and F O N L U P T [4], exept for the special case of B I X B Y ' S composition. The same holds for all further generalizations of the amalgam, e.g. the so-called 2-amalgam defined by C O R N U E J O L S and C U N N I N G H A M [7]
Finally, note t h a t a graph G generated from two graphs G i , G 2 G CQ by P * o p e r a t i o n s does not belong to LQ in general, as is shown by the two examples in Figure 4.
Figure 4: E x a m p l s for graphs genera
by P * o p e r a i o n s
et us now check, when complementaion, which preserves p e r f e c t e s s by LovÄsz's Perfect Graph Theorem [13], preserves critcal perfectness. Both a graph and ts complement are crit cally perfect if the deletion and the a d d i i o n of any edge results in an imperfect graph. This requirement is very strong. Indeed, the elements of CV have noncriical c o p l e m e n t s in general i e . , CV is not closed under c o p l e m e n t a t i o n . E.g., all linegraphs of ipartite graphs t h a t are not also H-graphs (see the linegraphs of th A g r a p h s shown in Figure 3), are n o c r i t i c a l
Obviously, he same holds for all graphs generate with n o c r i t i c a l complement
by P * o p e r a t i o n s fro
t l s t o e graph
However, there are many examples of c r i c a l l y perfect graphs with critical complements, e.g. the graphs show n Figure 1 and in Figure 4. We are able to prove some necessary conditions t h a t th c o p l e m e n t of a critically perfect graph is critical as well Lemma
2 If the complement
(i) is a connected
of a critically perfect graph G is critical, then G
graph,
(ii) has no cliquecutset v G E{G Vv G (ii) has no nonadjacent
t in at least two components
nodes with the same
ofG —
there exists a node x with
neighbourhood.
Proof. Let G be a critically perfect graph. (i) Assume t h a t G is disconnected with components G\ and G^ G, consider an edge e = x\X2 with x\ G V(G\), x G ViGq) and let Ge — e be the corresponding minimally imperfect subgraph of G — e. By LovÄsz [13], Ge + e C G + e is m i m a l l y imperfect as well, and e is the only edge connecting (Ge + e) n G\ and (Ge + e) n G- Therefore, Ge + e cannot be (2UJ — 2)connected, a contradiction to S E B Ö [18]. O (ii) Assume t h a Q is a cliquecutset of G, G\ and i are two c o p o n e n t s of G — Q, and x\ G G i , X2 G G are nodes with x\v, x^v G E(G) \/v G Q. In G, consider the edge e = X\X2 and the minimally imperfect subgraph Ge — e. By [13], Ge + e C G + e is minimally imperfect. Bu G e + e fl (Q U {x\ X) is a cliquecutset of Ge + e, a contradictio to C H V A T A L ' S Star-Cutset Lemma [5]. O (ii) Assum t h a t x\ and X2 are nonadjacent nodes with the same neighbourhood; in G, they are adjacent nodes and would dominate each other, a contradiction. • Therefore, either the c o p l e m e n t s of graphs generated by the unio (see emma 4.2(i)) no by the identificatio in a node or an edge (see Lemma 4.2(ii)) nor by the multiplicatio (see Lemma 4.2(ii)) are critically perfect. Furthermore, it follows by emma 4.2(i) t h a t th ellknown c o p l e t e join does not preserve critical p e r f e c t e s s We are further interested in - sufficient conditions t h a t guarantee the critical p e r f e c t e s s of a raph and ts c o p l e m e n t critically perfect graphs t h a t cannot be c o n s r u c t e d fro P*operations o coplementation,
the graphs in CQ by applying
- additional examples for P * o p e r a t i o n s , and - further operations preserving o c o n s r u c t i n g critical p e r f e c t e s s
Ackowedgent The autho is grateful to Prof. M. Grötschel for his suggestio perfect graphs and for his valuabl remarks
to investigate th class of critically
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