On the Classification Problem for Split Graphs

J Braz Comput Soc (2012) 18:95–101 DOI 10.1007/s13173-011-0046-2

SI: GRAPHCLIQUES

On the classiﬁcation problem for split graphs

Sheila Morais de Almeida · Célia Picinin de Mello · Aurora Morgana

Received: 5 October 2011 / Accepted: 20 October 2011 / Published online: 9 November 2011 © The Brazilian Computer Society 2011

Abstract The Classification Problem is the problem of de- An easy lower bound for the chromatic index is the max- ciding whether a simple graph has chromatic index equal to imum vertex degree . A celebrated theorem of Vizing [27] or + 1. In the first case, the graphs are called Class 1, states that, for a simple graph, the chromatic index is at most otherwise, they are Class 2.Asplit graph is a graph whose + 1. It was the origin of the Classification Problem that vertex set admits a partition into a stable set and a clique. consists of deciding whether a given graph has chromatic Split graphs are a subclass of chordal graphs. Figueiredo at index equal to or + 1. Graphs whose chromatic index al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) is equal to are said to be Class 1; graphs whose chro- state that a chordal graph is Class 2 if and only if it is matic index is equal to + 1 are said to be Class 2. De- neighborhood-overfull. In this paper, we give a characteri- spite the restriction imposed by Vizing, it is NP-complete to zation of neighborhood-overfull split graphs and we show determine, in general, if a graph is Class 1 [15]. In 1991, that the above conjecture is true for some split graphs. Cai and Ellis [2] proved that this holds also when the problem is restricted to some classes of graphs such as per- · · · Keywords Edge-coloring Overfull graph Split graph fect graphs. However, the classification problem is entirely Classification problem solved for some well-known classes of graphs that include the complete graphs, bipartite graphs [16], complete multipartite graphs [14], and graphs with universal vertices [20]. 1 Introduction Nevertheless, the complexity of the classification problem is unknown for several well-studied strongly structured graph An edge-coloring of G is an assignment of one color to each classes such as cographs [1], join graphs [17, 24, 25], pla- edge of G such that no adjacent edges have the same color. nar graphs [23], chordal graphs, and several subclasses of The chromatic index, χ(G), is the minimum number of col- chordal graphs such as split graphs [3], indifference graphs, ors for which G has an edge-coloring. interval graphs, and doubly chordal graphs [9]. By Vizing’s theorem, to show that a graph G is Class 1 S. Morais de Almeida () · C. Picinin de Mello it is enough to construct an edge-coloring for G with (G) Institute of Computing, University of Campinas, Campinas, Brazil colors, however, to show that G is Class 2 we must prove e-mail: [email protected] that G does not have an edge-coloring with (G) colors. C. Picinin de Mello Considering a simple graph G, the inequality |E(G)| > e-mail: [email protected] |V(G)| (G) 2 is a useful sufficient condition to classify G as a Class 2 graph. In such a way, this condition implies that S. Morais de Almeida Campus of Ponta Porã, Federal University of Mato Grosso do Sul, G has “many edges” and it is called overfull graph.Note Ponta Porã, Brazil that if a graph G is overfull, then G has an odd number |V(G)| of vertices and, since at most 2 edges of G can be A. Morgana colored with the same color, it is Class 2. Moreover, if a Department of Mathematics, University of Rome “La Sapienza”, = Roma, Italy graph G has an overfull subgraph H with (H ) (G), e-mail: [email protected] it is a subgraph-overfull graph [11]. When the overfull sub- 96 J Braz Comput Soc (2012) 18:95–101 graph H is induced by a (G)-vertex v and all its neighbors, The semicore of a graph G is the subgraph induced by denoted by N[v], we say that G is a neighborhood-overfull the core of G and their neighbors. An interesting result says graph [8]. Overfull, subgraph-overfull, and neighborhood- that the chromatic index of a graph is equal to the chromatic overfull graphs are Class 2. Although very rare, there are ex- index of its semicore [17]. In general, to solve the Classifi- amples of Class 2 graphs that are neither subgraph-overfull cation Problem for the semicore of G is as hard as to solve ∗ nor neighborhood-overfull. The smallest one is P ,the the Classification Problem for G. However, under special graph obtained from the Petersen graph by removing an ar- conditions of the semicore, a useful tool to solve the Classi- bitrary vertex. fication Problem for G is to solve it for its semicore. We use Hilton and Chetwynd [4] conjectured that being Class 2 this approach to classify some split graphs. is equivalent to being subgraph-overfull, when the graph In Sect. 2, we recall some known results that we use in the |V(G)| has a maximum degree greater than 3 . This conjec- subsequent sections. In Sect. 3, we give a characterization of ture is known as the Overfull Conjecture.EveryClass2 neighborhood-overfull split graphs, and in Sect. 4,weshow graph with maximum degree at least |V(G)|−3 is subgraph- that Conjecture 1 is true for some subclasses of split graphs. overfull [5]; every Class 2 complete multipartite graph is overfull [14]. These classes provide evidence for the Over- full Conjecture. Note that if the Overfull Conjecture is true, ∗ 2 Theoretical framework |V(P )| = the resulting theorem can not be improved, since 3 (P ∗) . In this paper, G denotes a simple, finite, undirected, and con- A split graph is a graph whose vertex set admits a par- nected graph with vertex set V(G)and edge set E(G). Write tition into a clique and a stable set. Split graphs are a n =|V(G)| and m =|E(G)|. A subgraph H of a graph well-studied class of graphs for which most combinato- G is a graph with V(H)⊆ V(G) and E(H) ⊆ E(G).For rial problems are solved [6, 7, 18, 19, 22]. It has been X ⊆ V(G), denote by G[X] the subgraph induced by X, proved that every overfull split graph contains a universal that is, V(G[X]) = X and E(G[X]) consists of those edges vertex and, therefore, is neighborhood-overfull. Moreover, of E(G) having both ends in X.LetD ⊆ E(G). The sub- every subgraph-overfull split graph is in fact neighborhood- graph induced by D is the subgraph H with E(H) = D and overfull [8]. In the same article, the authors have posed the V(H)is the set of every vertex of G with at least one edge following conjecture for chordal graphs (graphs without in- of D incident to it. For any v in V(G), the set of vertices duced cycles C with n ≥ 4), a superclass of split graphs. n adjacent to v is denoted by N(v) and N[v]={v}∪N(v). [ ] Conjecture 1 Every Class 2 chordal graph is neigh- The subgraph induced by N(v) and N v are called neighborhood-overfull. borhood of v and closed neighborhood of v, respectively. Two vertices, u and v, of a graph G are twin vertices if [ ]= [ ] ⊆ = Note that the validity of this conjecture for chordal graphs N u N v in G.ForX V(G), N(X) v∈X N(v). =| | and, therefore, for split graphs implies that the edge-coloring The degree of a vertex v is dG(v) N(v) .Themaximum = { : ∈ } problem for the corresponding class is in P . degree of G is (G) max dG(v) v V(G) .A(G)- In this work, we present a structural characterization of vertex is a vertex v with dG(v) = (G). When there is no the neighborhood-overfull split graphs. If Conjecture 1 is ambiguity, we remove the symbol G from the notation. true for split graphs, we are presenting a structural charac- A clique is a set of pairwise adjacent vertices of a graph. terization of the unique Class 2 split graphs. A maximal clique is a clique that is not properly contained in The study of the core and the semi-core of a graph gives any other clique. A stable set is a set of pairwise nonadjacent us some information about the Classification Problem. The vertices. A split graph G ={Q, S} is a graph whose vertex { } core of a graph G, denoted by G, is the subgraph of G in- set admits a partition Q, S into a clique Q and a stable duced by the (G)-vertices. The core of a graph has been set S. studied, since 1965, when Vizing [28] proved that G is In the following, we shall use some known results that Class 1 if G has at most two vertices. This result was later we recall for reader’s convenience. generalized by Fournier [10]: if G is a forest, then G is Class 1. Thus, the question was what happens when the core Theorem 2 [20] Let G be a graph with (G) =|V(G)| − | |≥ (G) of a graph contains a cycle. Hilton and Zhao [12, 13] con- 1. Then G is Class 1 if and only if E(G) 2 . sidered the graphs whose core is the disjoint union of cycles and paths, i.e., (G) = 2 and they conjectured that every Theorem 3 [17] The chromatic index of a graph G is equal ∗ graph with (G) = 2, different from P ,isClass2ifand to the chromatic index of its semicore. only if it is overfull. Tan and Hung [26] proved that this conjecture is true for split graphs. Note that a split graph with Theorem 4 [3] Let G ={Q, S} be a split graph. If (G) (G) = 2 has 3 vertices with maximum degree. is odd, then G is Class 1. J Braz Comput Soc (2012) 18:95–101 97

Theorem 5 [8] Let G ={Q, S} be a split graph. If G Theorem 9 Let G ={Q, S} be a split graph. The graph G is is overfull, then G has a universal vertex. Moreover, G is neighborhood-overfull if and only if the following conditions subgraph-overfull if and only if G is neighborhood-overfull. hold: 1. (G) is even; and Theorem 6 [26] Let G ={Q, S} be a split graph. Let X 2. there exists a set X ⊆ Q with at least k =|Q|− (G) + be the set of (G)-vertices. If |N(A) ∩ S|≥|A|, for each 2 d(Q) +1 (G)-vertices that are twins and, for a v ∈ X, A ⊆ X, then G is Class 1. 2 the number of edges of G[N[v]] incident to vertices of \ | |− Theorem 7 [26] Let G ={Q, S} be a split graph with Q X is at most Q k. (G) ≤ 2. Then G is Class 2 if and only if G is overfull. Proof Let G ={Q, S} be a split graph. Suppose that G is neighborhood-overfull. If G is a complete graph, every vertex is a (G)-vertex 3 A Class 2 split graph and all the conditions are trivially true. Therefore, we consider the case S = ∅. Since G is neighborhood-overfull, By Theorem 5, neighborhood-overfull and subgraph-over- G contains a (G)-vertex v such that G[N[v]] is overfull concepts are equivalent when restricts to split graphs. full. Then condition (1) is true. Moreover, by Theorem 2, In this section, we give a structural characterization of split |E(G[N[v]])| < (G) . So, there are at most (G) − 1 − graphs that are neighborhood-overfull. As far as we know, 2 2 d(Q) vertices in Q which are not adjacent to at least these graphs are the unique known Class 2 split graphs. 2 [ ]∩ [ [ ]] From now on, we consider a split graph G ={Q, S}, one vertex in N v S. Therefore, G N v contains at least k =|Q|− (G) + d(Q) + 1 vertices of maximum where Q is a maximal clique and S is a stable set. We 2 2 X shall associate to G a bipartite graph B obtained from G degree. Let be the set of the vertices of maximum degree in G[N[v]] (|X|≥k). Since v is a (G)-vertex, by removing all edges of the subgraph of G induced by Q. (G) = (G[N[v]]). Since |N[v]∩S|=d(Q), all vertices Let d(Q) be the maximum degree of a vertex of Q in the in X are (G)-vertices and they are twins. Furthermore, bipartite graph B, i.e., d(Q) = max{dB (v) : v ∈ Q}. Then d (w) ≤ (G) − 1 − d(Q) =|Q|−k. (G) =|Q|−1 + d(Q). w∈Q\X G[N[v]] 2 2 Now suppose that conditions (1) and (2) are true. Let v be one of the k vertices of maximum degree that are Lemma 8 Let G ={Q, S} be a split graph. If G is a twins and consider G[N[v]]. By condition (2), we have neighborhood-overfull graph, then |Q| and d(Q) must have |E(G[N[v]])|≤ d(Q) +|Q|−k = (G) − 1. By condi- different parities and |Q|=(d(Q))2 + i with i odd, i ≥ 3. 2 2 tion (1), G has even maximum degree. Therefore, G is a neighborhood-overfull graph. Proof Let G ={Q, S} be a split graph. If (G) is odd, by Theorem 4, G is Class 1 and, therefore, G is not The split graphs described in Theorem 9 are Class 2. neighborhood-overfull. Hence, (G) =|Q|+d(Q) − 1 Therefore, if the Conjecture 1 were true, these graphs would must be even. This implies that |Q| and d(Q) have differ- be the unique Class 2 split graphs and every split graph ent parities. G ={Q, S} with (G) even and |Q| <(d(Q))2 + 3 would Assume that G is neighborhood-overfull. If G is a com- be Class 1. plete graph, it is known that |Q| must be odd with |Q|≥3 = ∅ and the lemma follows. Therefore, we consider S . Corollary 10 Let G ={Q, S} be a neighborhood-overfull | | Since G is neighborhood-overfull, G contains a (G)- split graph. Then (G) > V(G) . vertex v such that G[N[v]] is overfull. Hence, by The- 3 | [ [ ]] |≤ (G) − orem 2, E(G N v ) 2 1. Since Q is a maxi- Proof Let G ={Q, S} be a neighborhood-overfull split ∈ [ ]∩ mal clique, for each u N v S there exists at least one graph. By Theorem 9, G has a set X of twin (G)-vertices vertex w ∈ Q such that {u, w} ∈E(G). Then d(Q) + 2 such that |NG(X) ∩ S|=d(Q) . Moreover, by Theorem 9, d(Q) ≤|E(G[N[v]])|≤ (G) − 1. This implies that |Q|≥ there are at most ( (G) −1)− d(Q) edges in G with an end 2 2 2 2 (d(Q)) + 3. Moreover, the parities of |Q| and d(Q) imply in Q \ X and another one in NG(X) ∩ S. Note that for each that |Q|=(d(Q))2 + i with i odd, i ≥ 3. one of these edges could exist an edge in G with an endpoint in Q \ X and another one in S \ N(X). We call byE this set (G) − − d(Q) Now we give a characterization of neighborhood-overfull of edges of G. So, there are at most ( 2 1) 2 ver- split graphs. It is relevant to note that the next theorem guar- tices of S that are endpoints of edges of E. Note that each antees that every neighborhood-overfull split graph G con- vertex of S belongs to N(X) or it is endpoint of an edge tains a minimum number of (G)-vertices that have the of E. Note also that the cardinality of S increases if the ver- same neighborhood. tices that are endpoints of edges of E are all distinct. So, the 98 J Braz Comput Soc (2012) 18:95–101 maximum cardinality of S is d(Q) + ( (G) − 1) − d(Q) , |X|−2, d(Q) ≥|X|−1. A (G)-vertex is also a (H )- 2 2 where d(Q) is the number of vertices belonging to N(X) vertex, then d(Q) ≥|X|−1. By hypothesis, H has no uni- and ( (G) − 1) − d(Q) is the maximum number of ver- v v 2 2 versal vertex, so there exist two distinct vertices i , j in \ = | ∩ |≥ tices belonging to S N(X) which are endpoints of edges X such that NH (vi) NH (vj ). Therefore, N(X) S | |≤ + (G) − − d(Q) d(Q)+ 1. Since d(Q)+ 1 ≥|X|, |N(X)∩ S|≥|X|. of E . Since S d(Q) ( 2 1) 2 ,wehave | |=| |+| |≤| |+ − + (G) − d(Q) Since d(Q) ≥|X|−1, if |A|=|X|, |N(A) ∩ S|= V(G) Q S Q d(Q) 1 2 . 2 |N(X)∩ S|≥|X|, and, for each proper subset A ⊂ X such Recall that (G) =|Q|+d(Q) − 1. So, V(G)≤(G) + (G) |V(G)| | |≤| |− | ∩ |≥| |− − d(Q) . Therefore, (G) ≥ 2( ) + 2 d(Q) > that A X 1, N(A) S X 1. Therefore, the 2 2 3 3 2 |V(G)| lemma follows. 3 . The next theorem shows that the Classification Problem The split graphs described in Theorem 9 are Class 2. is also solved for split graphs G that have (G) even and Therefore, if the Conjecture 1 were true, these graphs would (G) ≥|Q|+|X|−2, where X is the set of (G)-vertices. be the unique Class 2 split graphs and every split graph G ={Q, S} with (G) even and |Q| <(d(Q))2 + 3 would Theorem 13 Let G ={Q, S} be a split graph. Let X be the be Class 1. set of (G)-vertices. If (G) ≥|Q|+|X|−2, then G is Class 2 if and only if G is neighborhood-overfull. 4 Some Class 1 split graphs Proof Let G ={Q, S} be a split graph, let H be the semicore of G and let X be the set of (G)-vertices. In this section, we use the semi-core of a given split graph If H has a universal vertex, then by Theorem 2, G is G to determine the chromatic index of G. In general, find- Class 2 if and only if H is overfull. If H is overfull, by def- ing the chromatic index of the semicore of a graph G may inition of neighborhood-overfull graph, G is neighborhood- be as difficult as finding the chromatic index of G.IfG has the hereditary property for induced subgraphs, G and overfull. H its semicore belongs to the same class. This is the case Now consider without universal vertices. By the hy- (G) ≥|Q|+|X|− of split graphs. However, under special conditions this ap- pothesis, 2. Then, by Lemma 12 and proach may be useful for classifying certain split graphs. Theorem 6, H is Class 1. By Theorem 3, G is Class 1. Therefore, G is Class 2 if and only if G is neighborhood- Theorem 11 Let G ={Q, S} be a split graph and let H be overfull. the semicore of G. If H has a universal vertex, then G is ={ } | |≥ Class 2 if and only if H is overfull. Now, we consider a split graph G Q, S with X 4 vertices of maximum degree and an even (G) =|Q|+ | |− ≥ Proof Let G ={Q, S} be a split graph and let H be the X 3. Note that (G) 3. (Recall that Tan and = semicore of G. Then H is also a split graph and (H ) = Hung [26] proved that when (G) 2, a split graph G (G). To calculate the chromatic index of G, by Theorem 3, is Class 2 if and only if G is overfull.) First, we show that, ≥ =| |+| |− it is sufficient to calculate the chromatic index of its semi- when (G) 3 and (G) Q X 3, G is not | | core. By the hypothesis, H has a universal vertex. So, by neighborhood-overfull. After we show that, if X is odd, Theorem 2, H is Class 2 if and only if H is overfull. There- then G is Class 1. fore, G is Class 2 if and only if H is overfull. Lemma 14 Let G ={Q, S} be a split graph with even maxi- | |≥ By Theorem 11, the Classification Problem is solved for mum degree. Let X be the set of (G)-vertices with X 4. =| |+| |− a split graph if its semicore has a universal vertex. Now, we If (G) Q X 3, then G is not neighborhood- consider split graphs whose semicore has no universal ver- overfull. tex. Proof Let G ={Q, S} be a split graph with |X|≥4 vertices Lemma 12 Let G ={Q, S} be a split graph. Let X be the of maximum degree and let (G) =|Q|+|X|−3bean set of (G)-vertices. Let H be the semicore of G and sup- even number. Then d(Q) =|X|−2. If G is neighborhood- 2 pose that H does not contain universal vertices. If (G) ≥ overfull, then by Lemma 8, the condition |Q|≥(d(Q)) +3 2 |Q|+|X|−2, then for each A ⊆ X, |N(A)∩ S|≥|A|. implies that |Q|≥(|X|−2) + 3 and, by Theorem 9,the condition |Q|− (G) + d(Q) + 1 ≤|X| implies |Q|≤ 2 2 Proof Let H be the semicore of a split graph G ={Q, S} −|X|2 +8|X|−11. These inequalities imply that there exists and let X be the set of (G)-vertices. Then H is a split a neighborhood-overfull split graph if and only if |X|=3. graph with partition {Q, N(X) ∩ S}. Since (G) ≥|Q|+ Therefore, G is not neighborhood-overfull. J Braz Comput Soc (2012) 18:95–101 99

Lemma 15 Let G ={Q, S} be a split graph with even maxi- is the last one. Let U = (u0,u1,...,uh), where h is the mum degree. Let X be the set of (G)-vertices with |X| > 4. minimum number with |N(U)|≥|Q|/2. Let B be the bi- If (G) =|Q|+|X|−3 and |X| is odd, then G is Class 1. partite subgraph of L with partition {Q, U} and edges with an end in Q and another one in U. A CFK-ordering is an ={ } Proof Let G Q, S be a split graph with even maxi- ordering of the vertices of Q such that dB (vi) ≥ dB (vi+1) mum degree. Let X be the set of (G)-vertices with odd for each i = 0,...,|Q|−1. Moreover, if dB (vi) = dB (vj ), | | =| |+| |− =| |− X > 4. Since (G) Q X 3, d(Q) X 2. viuh ∈ E(L), and vj uh ∈ E(L), then i>j. By Lemma 14, G is not neighborhood-overfull and we shall We show that there exists a CFK-ordering of the vertices prove that G is Class 1. of Q such that y and z are the ﬁrst ones in the ordering. To calculate the chromatic index of G, by Lemma 3,it If |NB (y)| < |U|=h + 1, then dB (y) = dB (z) = dB (Q), is sufﬁcient to calculate the chromatic index of its semicore. where dB (Q) is the maximum degree of vertex of Q in the ={ } Let X v0,...,v|X|−1 be the set of (G)-vertices. Since subgraph B. Hence, dB (v) ≤ dB (y) for each v ∈ Q. More- ⊆ Q is maximal, X Q. Consider the semicore H of G.By over, if |NB (y)|

–Ifv|Q|−1 is a -vertex, then π(vw) is Acknowledgements We are grateful to anonymous referees for ⎧ their careful reading and valuable suggestions, which helped im- prove an earlier version of this note. This research was partially sup- ⎪π (vw), if vw ∈ M and vw = v v|Q|− ; ⎪ 0 1 ported by CAPES and CNPq (140709/2008-8, 482521/2007-4, and ⎪ ⎨⎪ − 1, if vw ∈ M and 473867/2010-9). The ﬁrst author was also partially supported by = = ; PROPP/UFMS. ⎪ vw v0u and vw v|Q|−1u ⎪ ⎪ − 1, if vw = v v| |− ; ⎩⎪ 0 Q 1 | |− = = Q 1, if vw v0u and vw v|Q|−1u . References If v|Q|−1 is not a -vertex, then the set of edges (M \ 1. Barbosa MM, de Mello CP, Meidanis J (1998) Local conditions {v0u}) ∪{v0v|Q|−1} is a matching, because M ∪{v1u} is a for edge-colouring of cographs. Congr Numer 133:45–55 matching. Otherwise, since X ⊆ Q and =|Q|+|X|−3 2. Cai L, Ellis JL (1991) NP-completeness of edge-colouring some restricted graphs. 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