Mutual placement of bipartite graphs - Semantic Scholar
Discrete Mathematics North-Holland
85
121 (1993) 85-92
Mutual placement of bipartite graphs* Jean-Luc
Fouquet
Laboratoire de Recherche en Irzformatique, 91405 Orsay Cedex, France
CIA 410 du C.N.R.S.,
Uniwrsite
de Paris-&d,
Bat. 490,
A. Pawel Wojda* * Departement
de Mathematiques
Received 31 October Revised 14 February
et Irzformatique. Faculte des Sciences, 72017 Le Mans Cedex, France
1990 1991
Abstract Fouquet, J.-L. and A.P. Wojda, (1993) 85-92.
Mutual
placement
of bipartite
graphs,
Discrete
Mathematics
121
Let G =(L, R, E) and H =(L’, R’, E’) be bipartite graphs. A bijection 4: L w R + L’v R’ is said to be of a biplacement of G and H if 4(L)= L’ and ~$(x)~$(y)gE’ for every edge xy of G. A biplacement G and its copy is called a 2-placement of G. We prove that, with some exceptions, every bipartite graph G of order n and size at most n - 2 has a 2-placement. We also give some sufficient conditions for bipartite graphs G and H to have a biplacement.
1. Introduction
Let G = (L, R, E) be a bipartite graph with a vertex set V(G) = L u R and an edge set E. We denote then L(G) = L, R(G) = R, and we call these sets the left and the right set of bipartition of the vertex set of G. The cardinality of the vertex set of G is called the order of G and is denoted by u(G), while the cardinality of the edge set E(G) = E is the size of G, denoted by e(G). For a vertex XEV(G), N(x, G) denotes the set of its neighbors in G. The degree d(x, G) of the vertex x in G is the cardinal of the set N(x, G); d,(G) and d,(G) are maximum vertex degree in the set L(G) and R(G), respectively. A vertex x of G is said to be pendent if d(x, G)= 1. Correspondence to: A. Pawel Wojda, Instytut Matematyki, Akademia Gorniczo-Hutnicza, Al. Mickiewicza 30, 30-059 Krakow, Poland. * Partially supported by Polish Research Grant KBE p/05/003/90-2. **On leave from lnstytut Matematyki, Akademia Gorniczo-Hutnicza, Al. Mickiewicza 30, 30-059 Krakow, Poland. 0012-365X/93/$06.00
6
1993-Elsevier
Science Publishers
B.V. All rights reserved
86
J.-L. Fouquet. A.P.
Wojdu
K,, stands for the complete bipartite graph with )L(K,,)I =p and IR(K,,)( =q. If JLJ =p and ]RJ =q, we say that G is a (p,q)-bipartite graph. Then Gbip is the complement joining
of G in K,,.
Gbip =(L,R,E’),
Thus
where
L with R which are not in E. For bipartite
G I”;H the vertex-disjoint
union
R(GI:H)=R(G)uR(H),
E(G:H)=E(G)uE(H)
R(H) are supposed
of graphs
to be mutually
E’ consists
graphs
G and H. Thus,
of all the edges
G and H, we denote
by
L(G ?J H) = L(G) u L(H),
and the sets L(G),R(G),L(H)
and
disjoint.
Let G=(L,R, E) and H=(L’,R’, E’) be two (p,q)-bipartite graphs. We say that G and H are mutually placeable (or just mp) if there is a bijection 4 : L u R + L’ u R’ such that
cj(L)=L’
and, for every edge xy~E,
cj(x)&(y)
is not an edge of H. The
function 4 is called the biplacement of G and H. A 2-placement of G is a biplacement of G and its copy. If such a 2-placement of G exists then we say that G is 2-placeable. Finding a biplacement of two (p, q)-bipartite graphs G and H is equivalent to finding in the graph K,,- G a copy of H. We then call the edges of G black and the edges of H red. A 2-placement of (p, q)-bipartite graph G is a red copy G, of Gb = G, such that L(G,)= L(G,), R(Gb)=R(G,) and E(G,)nE(G,)=tii Note that for two bipartite graphs G and H it may happen that for some bipartitions of the vertex sets they may be mp while they are not mp for some other bipartitions. Thus, if one asks if bipartite graphs G and H are mp then the left and right sets of bipartitions of the sets V(G) and V(H) must be made precise. Exhaustive surveys of the results concerning the problems of placing of graphs are given in [2, Chapter S] and [S, Chapter 41; however, the particular case of placement of bipartite graphs, in the sense of the definition given above, has probably not been considered yet. See also [3] for another approach to the placement of bipartite graphs. In this paper we give some sufficient conditions for bipartite graphs to be mp or 2-placeable.
2. 2-Placements In the main result of this section we shall prove that every (p,q)-bipartite graph (p, q 3 2) of size at most p + q - 2 is either 2-placable or its size is exactly p + q - 2 and G is in a family of graphs which is defined below. A (p, q)-bipartite graph of size q (p) is said to be left-side bistar SL(p, q) (right-side bistar SR(p, q)) if there is a vertex of degree q (p) in its left (right) set of bipartition. Note that if p 3 2 then two left-side stars are mp, while SL( p, q) and SR( p, q) are not mp. For p 3 1 and q >, 1 the (p, q)-bipartite graph BS(p, q) is the vertex-disjoint union of the left-side star SL(1, q - 1) and the right-side star SR(p - 1, l), as shown in Fig. la. The vertices denoted by x and y in Fig. la are called the centers of BS(p,q). Let r, s, t be nonnegative integers such that r+s+t>, 1. Then the graphs F = F(3, r, s, t) are defined as follows: F is a (3,2(r + s + t)- 1)-bipartite graph of size 2(r+s+t) such that L(F)={ a1,az,as}, R(F)=A,uAzuAsu& lA,l=r, lAzl=~, for i = 1,2,3 (mod 3) and the vertices of lA3(=t, IB(=r+S+t-1, N(Uf,F)=AiUAi+l
Mutual placement
a
. w :A
b
of bipartite graphs
87
r
B (3,r, s) (Cl
Hf3,r. sj (d) Fig. 1
B are isolated (see Fig. 2, where the vertices of L(F) are indicated by l , while o stand for the vertices of R(F)). For r and s nonnegative integers, r B 1, B = B(3, r, s) is (3,2r + s - 1)-bipartite graph ofsize2r+ssuchthatL(B)={a,b,c},R(B)=AuBuC,~AJ=r,~B(=s,(C~=v-1and xywithx~L(B),yVER(B)isanedgeofB(3,r,s)ifandonlyifx~{a,b}andy~Aorx=c and DEB (see Fig. lc). H=H(3, r, s) is the (3,2r+ 2s- 1)-bipartite graph of size 2r + 2s in which L(H)= {a, b, c}, while R(H) is the union of four mutually disjoint sets A, B, C and D with JAJ=ICJ=r, IBI=IDJ+l=s, N(a,H)=AuB, N(b,H)=BuC, while c is isolated vertex (see Fig. lc). We define Y(p,q) to be the set of (p,q)-bipartite graphs G such that either - p > 3, q > 3 and G is isomorphic to BS( p, q), or - p + q = 2r + 2s + 2t + 2, 3 E( p, q) and G is isomorphic to F(3, r, s, t), or - p+q=2r+s+2, 3c{p,q}, s32r and G is isomorphic to B(3,r,s), or else - p+q=2r+2s+2, 3~{p, q} and G is isomorphic to H(3,r, s).
J.-L. Fouquet, A.P. Wojda
88
It is not difficult to see that every graph in 9(p, q) is (p, q)-bipartite has no 2-placement.
of size p + q - 2 and
Theorem 2.1. Let G be a (p, q)-bipartite graph of size at most p + q - 2, p > 2, q > 2. Then either G is 2-placeable
To prove Theorem
or e(G)=p+q-2
and G~%(p,q).
1, we shall need two lemmas.
Lemma 2.2. Every (2, q)-bipartite graph G of size at most q has a 2-placement. Proof. We may suppose @=IN(a,G)-N(b,G)I,
that the size of G is equal to q. Let L(G)= P=IN(b,G)-N(a,G)l
q=a+P+2y=a+fi+y+
y=jl(
and
the
{a, b}, and put
and Y=IN(a,G)nN(b,G)J. We have I is the set of isolated vertices in R(G). Thus, G, in which N(a,G,)=(N(b,G)-N(a,G))ul and defines a 2-placement of G. 0
111, where
red
graph
N(b,G,)=(N(a,G)-N(b,G))ul
Lemma 2.3. Let p and q be two integers, p > 3, q > 3. Suppose that, for p’ > 2, q’ > 2 and p’+q’, 3, there is in L(G) a vertex v such that d(v, G) > 2. If the graph G’= G - {u, v} is 2-placeable then it is not difficult to prove that the same holds for G. If it is not the case then either p=3 or else d(v, G)=2 and G’~9(p-2,q). When p = 3, then one may show that G is either 2-placeable or isomorphic with H(3, r, s, t). By inspection of all possible cases, the reader may check that also when d(v, G)=2 and G’eY(p, q) then either G is 2-placeable or G~%(p,q). Case 2: There is no isolated vertex in G and there are two pendent vertices u and u such that u and v are in the same set of bipartition of V(G) and N (u, G) # N (v, G).
Let us denote by x the neighbor of u and by y the neighbor of v. We may suppose that u, VEL(G). Note that if p= 3 then there is only one graph of size p+q -2, without isolated vertex and with two pendent vertices u, VEL(G), N (u, G) #N (v, G) and, moreover, this graph has a 2-placement. Thus, suppose that p 2 4.
Mutual placement of‘ bipartite graphs
If G’= G- {u, O} has a 2-placement,
4’ say, then define a 2-placement
following way: - $(w)=~‘(w) for WEI’({u,u}; ~ if ~‘(x)#x and $‘(y)#y then ~(u)=u
all possible
cases.
of G in the
and c#J(u)=u;
- if either $‘(x)=x or 4’(y)=y then c$(u)=u and ~(u)=u. If G’ has no 2-placement then G’E%(p’,q’). The reader may conclude inspecting
89
the lemma
0
Now we may prove the main theorem
of this section.
Proof of Theorem 2.1 (by induction on p + q). By Lemma 2.2, the theorem holds when either p or q is equal to 2. So, we may suppose that ~33, q33 and that the theorem holds for every (p’, q’)-bipartite graph G’ of size at most p’ + q’ - 2 when p’ + q’
85
121 (1993) 85-92
Mutual placement of bipartite graphs* Jean-Luc
Fouquet
Laboratoire de Recherche en Irzformatique, 91405 Orsay Cedex, France
CIA 410 du C.N.R.S.,
Uniwrsite
de Paris-&d,
Bat. 490,
A. Pawel Wojda* * Departement
de Mathematiques
Received 31 October Revised 14 February
et Irzformatique. Faculte des Sciences, 72017 Le Mans Cedex, France
1990 1991
Abstract Fouquet, J.-L. and A.P. Wojda, (1993) 85-92.
Mutual
placement
of bipartite
graphs,
Discrete
Mathematics
121
Let G =(L, R, E) and H =(L’, R’, E’) be bipartite graphs. A bijection 4: L w R + L’v R’ is said to be of a biplacement of G and H if 4(L)= L’ and ~$(x)~$(y)gE’ for every edge xy of G. A biplacement G and its copy is called a 2-placement of G. We prove that, with some exceptions, every bipartite graph G of order n and size at most n - 2 has a 2-placement. We also give some sufficient conditions for bipartite graphs G and H to have a biplacement.
1. Introduction
Let G = (L, R, E) be a bipartite graph with a vertex set V(G) = L u R and an edge set E. We denote then L(G) = L, R(G) = R, and we call these sets the left and the right set of bipartition of the vertex set of G. The cardinality of the vertex set of G is called the order of G and is denoted by u(G), while the cardinality of the edge set E(G) = E is the size of G, denoted by e(G). For a vertex XEV(G), N(x, G) denotes the set of its neighbors in G. The degree d(x, G) of the vertex x in G is the cardinal of the set N(x, G); d,(G) and d,(G) are maximum vertex degree in the set L(G) and R(G), respectively. A vertex x of G is said to be pendent if d(x, G)= 1. Correspondence to: A. Pawel Wojda, Instytut Matematyki, Akademia Gorniczo-Hutnicza, Al. Mickiewicza 30, 30-059 Krakow, Poland. * Partially supported by Polish Research Grant KBE p/05/003/90-2. **On leave from lnstytut Matematyki, Akademia Gorniczo-Hutnicza, Al. Mickiewicza 30, 30-059 Krakow, Poland. 0012-365X/93/$06.00
6
1993-Elsevier
Science Publishers
B.V. All rights reserved
86
J.-L. Fouquet. A.P.
Wojdu
K,, stands for the complete bipartite graph with )L(K,,)I =p and IR(K,,)( =q. If JLJ =p and ]RJ =q, we say that G is a (p,q)-bipartite graph. Then Gbip is the complement joining
of G in K,,.
Gbip =(L,R,E’),
Thus
where
L with R which are not in E. For bipartite
G I”;H the vertex-disjoint
union
R(GI:H)=R(G)uR(H),
E(G:H)=E(G)uE(H)
R(H) are supposed
of graphs
to be mutually
E’ consists
graphs
G and H. Thus,
of all the edges
G and H, we denote
by
L(G ?J H) = L(G) u L(H),
and the sets L(G),R(G),L(H)
and
disjoint.
Let G=(L,R, E) and H=(L’,R’, E’) be two (p,q)-bipartite graphs. We say that G and H are mutually placeable (or just mp) if there is a bijection 4 : L u R + L’ u R’ such that
cj(L)=L’
and, for every edge xy~E,
cj(x)&(y)
is not an edge of H. The
function 4 is called the biplacement of G and H. A 2-placement of G is a biplacement of G and its copy. If such a 2-placement of G exists then we say that G is 2-placeable. Finding a biplacement of two (p, q)-bipartite graphs G and H is equivalent to finding in the graph K,,- G a copy of H. We then call the edges of G black and the edges of H red. A 2-placement of (p, q)-bipartite graph G is a red copy G, of Gb = G, such that L(G,)= L(G,), R(Gb)=R(G,) and E(G,)nE(G,)=tii Note that for two bipartite graphs G and H it may happen that for some bipartitions of the vertex sets they may be mp while they are not mp for some other bipartitions. Thus, if one asks if bipartite graphs G and H are mp then the left and right sets of bipartitions of the sets V(G) and V(H) must be made precise. Exhaustive surveys of the results concerning the problems of placing of graphs are given in [2, Chapter S] and [S, Chapter 41; however, the particular case of placement of bipartite graphs, in the sense of the definition given above, has probably not been considered yet. See also [3] for another approach to the placement of bipartite graphs. In this paper we give some sufficient conditions for bipartite graphs to be mp or 2-placeable.
2. 2-Placements In the main result of this section we shall prove that every (p,q)-bipartite graph (p, q 3 2) of size at most p + q - 2 is either 2-placable or its size is exactly p + q - 2 and G is in a family of graphs which is defined below. A (p, q)-bipartite graph of size q (p) is said to be left-side bistar SL(p, q) (right-side bistar SR(p, q)) if there is a vertex of degree q (p) in its left (right) set of bipartition. Note that if p 3 2 then two left-side stars are mp, while SL( p, q) and SR( p, q) are not mp. For p 3 1 and q >, 1 the (p, q)-bipartite graph BS(p, q) is the vertex-disjoint union of the left-side star SL(1, q - 1) and the right-side star SR(p - 1, l), as shown in Fig. la. The vertices denoted by x and y in Fig. la are called the centers of BS(p,q). Let r, s, t be nonnegative integers such that r+s+t>, 1. Then the graphs F = F(3, r, s, t) are defined as follows: F is a (3,2(r + s + t)- 1)-bipartite graph of size 2(r+s+t) such that L(F)={ a1,az,as}, R(F)=A,uAzuAsu& lA,l=r, lAzl=~, for i = 1,2,3 (mod 3) and the vertices of lA3(=t, IB(=r+S+t-1, N(Uf,F)=AiUAi+l
Mutual placement
a
. w :A
b
of bipartite graphs
87
r
B (3,r, s) (Cl
Hf3,r. sj (d) Fig. 1
B are isolated (see Fig. 2, where the vertices of L(F) are indicated by l , while o stand for the vertices of R(F)). For r and s nonnegative integers, r B 1, B = B(3, r, s) is (3,2r + s - 1)-bipartite graph ofsize2r+ssuchthatL(B)={a,b,c},R(B)=AuBuC,~AJ=r,~B(=s,(C~=v-1and xywithx~L(B),yVER(B)isanedgeofB(3,r,s)ifandonlyifx~{a,b}andy~Aorx=c and DEB (see Fig. lc). H=H(3, r, s) is the (3,2r+ 2s- 1)-bipartite graph of size 2r + 2s in which L(H)= {a, b, c}, while R(H) is the union of four mutually disjoint sets A, B, C and D with JAJ=ICJ=r, IBI=IDJ+l=s, N(a,H)=AuB, N(b,H)=BuC, while c is isolated vertex (see Fig. lc). We define Y(p,q) to be the set of (p,q)-bipartite graphs G such that either - p > 3, q > 3 and G is isomorphic to BS( p, q), or - p + q = 2r + 2s + 2t + 2, 3 E( p, q) and G is isomorphic to F(3, r, s, t), or - p+q=2r+s+2, 3c{p,q}, s32r and G is isomorphic to B(3,r,s), or else - p+q=2r+2s+2, 3~{p, q} and G is isomorphic to H(3,r, s).
J.-L. Fouquet, A.P. Wojda
88
It is not difficult to see that every graph in 9(p, q) is (p, q)-bipartite has no 2-placement.
of size p + q - 2 and
Theorem 2.1. Let G be a (p, q)-bipartite graph of size at most p + q - 2, p > 2, q > 2. Then either G is 2-placeable
To prove Theorem
or e(G)=p+q-2
and G~%(p,q).
1, we shall need two lemmas.
Lemma 2.2. Every (2, q)-bipartite graph G of size at most q has a 2-placement. Proof. We may suppose @=IN(a,G)-N(b,G)I,
that the size of G is equal to q. Let L(G)= P=IN(b,G)-N(a,G)l
q=a+P+2y=a+fi+y+
y=jl(
and
the
{a, b}, and put
and Y=IN(a,G)nN(b,G)J. We have I is the set of isolated vertices in R(G). Thus, G, in which N(a,G,)=(N(b,G)-N(a,G))ul and defines a 2-placement of G. 0
111, where
red
graph
N(b,G,)=(N(a,G)-N(b,G))ul
Lemma 2.3. Let p and q be two integers, p > 3, q > 3. Suppose that, for p’ > 2, q’ > 2 and p’+q’, 3, there is in L(G) a vertex v such that d(v, G) > 2. If the graph G’= G - {u, v} is 2-placeable then it is not difficult to prove that the same holds for G. If it is not the case then either p=3 or else d(v, G)=2 and G’~9(p-2,q). When p = 3, then one may show that G is either 2-placeable or isomorphic with H(3, r, s, t). By inspection of all possible cases, the reader may check that also when d(v, G)=2 and G’eY(p, q) then either G is 2-placeable or G~%(p,q). Case 2: There is no isolated vertex in G and there are two pendent vertices u and u such that u and v are in the same set of bipartition of V(G) and N (u, G) # N (v, G).
Let us denote by x the neighbor of u and by y the neighbor of v. We may suppose that u, VEL(G). Note that if p= 3 then there is only one graph of size p+q -2, without isolated vertex and with two pendent vertices u, VEL(G), N (u, G) #N (v, G) and, moreover, this graph has a 2-placement. Thus, suppose that p 2 4.
Mutual placement of‘ bipartite graphs
If G’= G- {u, O} has a 2-placement,
4’ say, then define a 2-placement
following way: - $(w)=~‘(w) for WEI’({u,u}; ~ if ~‘(x)#x and $‘(y)#y then ~(u)=u
all possible
cases.
of G in the
and c#J(u)=u;
- if either $‘(x)=x or 4’(y)=y then c$(u)=u and ~(u)=u. If G’ has no 2-placement then G’E%(p’,q’). The reader may conclude inspecting
89
the lemma
0
Now we may prove the main theorem
of this section.
Proof of Theorem 2.1 (by induction on p + q). By Lemma 2.2, the theorem holds when either p or q is equal to 2. So, we may suppose that ~33, q33 and that the theorem holds for every (p’, q’)-bipartite graph G’ of size at most p’ + q’ - 2 when p’ + q’
o(Hi) for i= 1, . . . . m, we have necessarily k 2 2. By Lemma 2.3, we may assume that there is no isolated vertex in G and that neither in L(G) nor in R(G) there are no two pendent vertices with different neighbors. Thus, k=2 and, moreover, Tl u T2 = BS(r, s), r, ~23, and there is no pendent vertex in uH,)=e(H,u...uH,) and H1,...,H, are vertexHlu...uH,. So, u(H,u... disjoint cycles. Let xeL(G) and PER be the centers of Tl u T,, and let UEL(H,) and bER(H,) be two consecutive vertices on the cycle HI. The graph H = HI u ... u H,-- {a, b) satisfies e(H) = u(H)- 3 and, by the induction hypothesis, H has a 2-placement, 4’ say. Then 4 defined by - +(cc)=~‘(N) for aelf({u,b,x,y}, -
4(4=x,
is a 2-placement
4(x)=u,
$(b)=y,
&y)=b
of G. This proves Theorem
3. B&placement of two (p,q)-bipartite
2.1.
0
graphs
Theorem 3.1. Let G and H be two (p, q)-bipartite graphs such that e(G)e(H)dpq. Then G and H are mp unless {G, H} = {F,, F,}, wh ere either F1 is the leji-side bistur and there is no isolated vertex in L(F,),
or F, is the right-side star and there is no isolated vertex
in R(F,).
Proof. Let G and H be two graphs satisfying the assumptions of the theorem. For ecE(G) and fEE(H), we shall denote by U(e,S) the set of all bijections o of L(G)uR(G) onto L(H)uR(H), such that a(L(G))=L(H) and a*(e)=& The number of all bijections CJof L(G)uR(G) onto L(H)uR(H) satisfying o(L(G))=L(H) and which are not a biplacement of G and H is then equal to the cardinality of the set
90
J.-L. Fouyuet.
U(e,f). u=u?&(G)UfcEW)
are
exactly
Wqjda
We then have
and since lU(e,f)I=(p-l)!(q-l)!, There
A.P.
we obtain
p!q!
bijections
lUlGp!q!.
o:L(G)uR(G)+L(H)uR(H)
satisfying
a(L(G) = L(H)). If 1U 1
