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Seidel complementation on (P _5, House, Bull)-free graphs
Jean-Luc Fouquet, Jean-Marie Vanherpe
To cite this version:
Jean-Luc Fouquet, Jean-Marie Vanherpe. Seidel complementation on (P _5,
House,Bull)-free graphs.2010. �hal-00467642�
J.L. FOUQUET AND J.M. VANHERPE Abstract.
1. Seidel complementation
Notation 1.1. N(v)denotes the neighborhood of the vertex v,N[v] =N(v)∪ {v}
andN(v) =V −N(v).
Given a graphG= (V, E) and a vertexv ofG, the seidel complement ofGonv inverses all edges betweenN(v) andV −N[v]. More formaly
Definition 1.2. Let = (V, E) be an undirected graph and let v be a vertex ofG.
The Seidel complement oatv onG, denotedG∗v is defined as follows :
G =v = (V, E1∪E2∪E3) where E1 ={xy|xy ∈E, x ∈ N[v], y ∈ N[v]}, E2 = E∩N(v)2 andE3={xy|xy /∈E, x∈N(v), y∈N(v)}.
Remark 1.3. [3]
• G∗v∗v=G.
• If Gis a cograph andv is a vertex ofGthen G∗v is a cograph.
• G is prime with respect to modular decomposition if and only if G∗v is prime with respect to modular decomposition
• G∗v=G∗v
2. seidel complementation on (P5,P5,Bull)-free graphs
Theorem 2.1. A graph Gis (P5,House,Bull)-free if and only if for all vertexv of G,G∗v is (P5,House,Bull)-free.
Proof Letvbe a vertex ofG. Assume thatG∗vcontains an induced subgraph, sayH,which is isomorphic to either aP5 or aHouse or aBull.
Claim 2.1.1. The vertex v does not belong toH.
1991Mathematics Subject Classification. 035 C.
P5 House Bull
Figure 1. Forbidden configurations for (P5, House,Bull)-free graphs.
1
2 J.L. FOUQUET AND J.M. VANHERPE
v N(v) N(v)
v
N(v) N(v)
v v
N(v) N(v) N(v) N(v)
v
N(v) N(v)
v N(v) N(v)
v N(v) N(v)
v
N(v) N(v)
v N(v) N(v)
Figure 2. The possible cases whenv is a vertex ofH.
Proof Assume not. Figure 2 describes all cases that can occur whenever v is a vertex of H. It is not difficult to check that, in all cases, (G∗v)∗v would contain a subgraph isomorphic toP5 or toHouseor toBull, a contradiction since G∗v∗v=GandGis assumed to be (P5,House,Bull)-free.
In the following we suppose that the vertices of H are in N(v)∪N(v), moreover H has vertices in both sets N(v) and N(v), otherwise H would be an induced subgraph ofG∗v∗v=G, a contradiction.
Figure 3 (resp Figure 4, Figure 5) describes all cases that can occur wheneverH is isomorphic to aP5(resp. aBull, aHouse) and has at most two vertices inN(v).
In all cases we get a contradiction with Claim 2.1.1 or a forbidden configuration appears inG∗v∗v. When H has more than 2 vertices in N(v) we get a similar
contradiction in consideringG.
3. seidel complementation on the modular decomposition tree of (P5, P5,Bull)-free graphs
3.1. On (P5, P5)-free graphs. We shall say that an induced subgraph of a (P5,(P5))-free graph is a buoy[1] whenever we can find a partition of its vertex set into 5 subsets Ai, i = 1, . . . ,5 (subsript i is to be taken modulo 5, such that Ai and Ai+1 are joined by every possible edge, while no possible edge are allowed betweenAi andAj whenj6=i+ 1mod[5], and such that theAi’s are maximal for these properties.
Theorem 3.1. [1, 2] Let G be a connected (P5,P5)-free graph. If G contains an induced C5 then every C5 of G is contained into a buoy, and this buoy is either equal to Gor is an homogeneous set ofG.
Corollary 3.2. Every prime (P5,P5)-free graph is either aC5 or isC5-free
N(v) N(v)
v v
N(v) N(v)
v
N(v) N(v) v
v
N(v) N(v)
v
N(v) N(v)
v
N(v) N(v)
v v
Figure 3. Cases : H is aP5 and has at most 2 vertices inN(v).
v
N(v) N(v)
v
N(v) N(v)
v v
N(v) N(v) v
N(v) N(v) N(v)
v
N(v)
v
N(v) N(v)
N(v) N(v)
v
N(v) N(v)
v
N(v) N(v)
Figure 4. Cases : H is aBull and has at most 2 vertices inN(v).
3.2. On (P5,P5,Bull)-free graphs.
Theorem 3.3. [1]Let Gbe a prime graph. Gis a P5HB-free graph if and only if one of the following conditions is satisfied
(i) Gis isomorphic to a C5; (ii) Gis bipartite andP5-free;
(iii) Gis bipartite andP5-free.
In a primeP5-free biparte graph the neighborhoods of two distinct vertices can- not overlap properly, thus :
Proposition 3.4. Let G = (V, E) be a prime graph of n vertices. G is bipartite andP5-free iff the following conditions are verified:
4 J.L. FOUQUET AND J.M. VANHERPE
v
v
N(v) N(v) N(v) N(v)
v v
N(v) N(v) N(v)
v
N(v)
v
N(v) N(v)
N(v) N(v)
v
N(v) N(v)
v
N(v) N(v)
Figure 5. Cases : H is a Houseand has at most 2 vertices inN(v).
b1 w5
w4
w3
w2
b5 b4 b3 b2
w1
Figure 6. A PrimeP5-free bipartite graph.
(i) There exists a partition of V(G) into two stable sets B = {b1, b2, . . . bn2} andW ={w1, w2, . . . wn2}.
(ii) The neighbors ofbi (i= 1. . .n2) are precisely w1, . . . wn2−i+1.
3.3. Seidel complementation of a Prime P5-free bipartite graph.
Proposition 3.5. LetG= (B∪W, E)be a prime bipartite P5-free graph such that B ={b1, . . . bn2} and W ={w1, . . . , wn2}, then G∗bi is a prime P5-free bipartite graph together with the bipartition :
B′={bi−1, bi−2, . . . bi, w1, w2, . . . , wn2}, W′={bi+1, . . . , bn2, wn2, wn2−1, . . . wn2−i+1}
Proof We haveN(bi) ={wn2, wn2−1, . . . wn2−i+1}andN(bi) ={b1, b2, . . . , bi−1, wn2−i, . . . , w1}.
It must be pointed out that the subgraphs induced by the setsS1={wn2, wn2−1, . . . wn2−i+1, b1, b2, . . . , bi−1} andS2={bi+1, . . . , bn2, wn2−i, . . . w1}are primeP5-free bipartite graphs as well as
S3={wn2, wn2−1, . . . wn2−i+1, bi+1, . . . , bn2}induces a complete bipartite graph. The result follows when considering the Seidel complementation onbi(see Figure 7).
Corollary 3.6. For a prime P5-free bipartite graph (or its complement) Seidel complementation at any vertex can be performed in constant time.
w2
b n 2 b n
2
G∗bi G∗bi
G
bi+1 b n
2−1 b n
2
w n 2−2 bi−2
−1
b1
w n 2−i
b n 2−1
1 b2
bi
2 w n
2−1
w n 2−i+1
w1 w2
bi w n
2 2
w1
w2
G
bi−1 1 b2
bi w n
2−1
w n 2−i+1 w n
2−i+1 w n
2−1
bi
b2 1
bi−1
w2
w1 bi+1 w n
2−i
b n 2−1
w n 2−i
bi+1
w1
b4
b5
Figure 7. Seidel complementation onb2.
References
[1] J. L. Fouquet. A decomposition for a class of (P5, P5)-free graphs. Dicrete Mathematics, 121:75–83, 1993.
[2] J.L. Fouquet, V. Giakoumakis, H. Thuillier, and F. Maire. On graphs withoutP5 and P5. Discrete Mathematics, 146:33–44, 1995.
[3] V. Limouzy. Seidel minor, permutation graphs and combinatorial properties (extended ab- stract. Technical report.
L.I.F.O., Facult´e des Sciences, B.P. 6759, Universit´e d’Orl´eans, 45067 Orl´eans Cedex 2, FR
