Claw-free Graphs Research Articles

Polynomials with Real Zeros and Compatible Sequences

In this paper, we study polynomials with only real zeros based on the method of compatible zeros. We obtain a necessary and sufficient condition for the compatible property of two polynomials whose leading coefficients have opposite sign. As applications, we partially answer a question proposed by M. Chudnovsky and P. Seymour in the recent publication [M. Chudnovsky, P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B 97 (2007) 350--357]. We also establish the connection between the interlacing property and the compatible property of two polynomials and give a simple proof of some known results.

Paired-Domination in Claw-Free Graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number, denoted by ? pr (G), is the minimum cardinality of a paired-dominating set in G. In this paper we investigate the paired-domination number in claw-free graphs. Specifically, we show that ? pr (G) ≤ (3n ? 1)/5 if G is a connected claw-free graph of order n with minimum degree at least three and that this bound is sharp.

Claw-free graphs. VII. Quasi-line graphs

A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. Here we give a construction for all quasi-line graphs; it turns out that there are basically two kinds of connected quasi-line graphs, one a generalization of line graphs, and the other a subclass of circular arc graphs.

Circumferences of 2-factors in claw-free graphs

A graph G is said to be claw-free if G has no induced subgraph isomorphic to K1,3. Matthews and Sumner [M.M. Matthews, D.P. Sumner, Longest paths and cycles in K1,3-free graphs, J. Graph Theory 9 (1985) 269–277] proved that c(G)≥min{2δ(G)+4,n} if G is a 2-connected claw-free graph of order n, where for a graph G, let δ(G) and c(G) denote the minimum degree and the length of a longest cycle of G, respectively. In this paper, we extend this result for a graph G with δ(G)≥7 as follows: if G is a 2-connected claw-free graph of order n with δ(G)≥7, then G has a 2-factor F such that c(F)≥min{2δ(G)+4,n}.

Stability of Hereditary Graph Classes Under Closure Operations

If is a subclass of the class of claw-free graphs, then is said to be stable if, for any , the local completion of G at any vertex is also in . If is a closure operation that turns a claw-free graph into a line graph by a series of local completions and is stable, then for any . In this article, we study stability of hereditary classes of claw-free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.

Hamiltonian Connectedness in Claw-Free Graphs

If every vertex cut of a graph G contains a locally 2-connected vertex, then G is quasilocally 2-connected. In this paper, we prove that every connected quasilocally 2-connected claw-free graph is Hamilton-connected.

On 2-factors in claw-free graphs whose edges are in small cycles

It is showed that every simple claw-free graph of minimum degree at least 3 in which every edge lies in a cycle of length at most 5 has a 2-factor.

Locating-total domination in claw-free cubic graphs

In this paper, we continue the study of locating-total domination in graphs. A set S of vertices of a graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. We consider total dominating sets S which have the additional property that distinct vertices in V(G)∖S are totally dominated by distinct subsets of the total dominating set. Such a set S is called a locating-total dominating set in G, and the locating-total domination number of G is the minimum cardinality of a locating-total dominating set in G. A claw-free graph is a graph that does not contain K1,3 as an induced subgraph. We show that the locating-total domination number of a claw-free cubic graph is at most one-half its order and we characterize the graphs achieving this bound.

Fast exact algorithm for [formula omitted]-labeling of graphs

An L(2,1)-labeling of a graph is a mapping from its vertex set into nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. The span of such a labeling is the maximum label used, and the L(2,1)-span of a graph is the minimum possible span of its L(2,1)-labelings. We show how to compute the L(2,1)-span of a connected graph in time O∗(2.6488n). Previously published exact exponential time algorithms were gradually improving the base of the exponential function from 4 to the so far best known 3, with 3 itself seemingly having been the Holy Grail for quite a while. As concerns special graph classes, we are able to solve the problem in time O∗(2.5944n) for claw-free graphs, and in time O∗(2n−r(2+nr)r) for graphs having a dominating set of size r.

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G − V (S). For an integer k ≥ 4, a graph G has the single k-cycle property if every edge of G, which does not lie in a triangle, lies in a cycle C of order at most k such that C has at least \(\left\lfloor {\tfrac{{\left| {V(C)} \right|}} {2}} \right\rfloor \) edges which do not lie in a triangle, and they are not adjacent. In this paper, we show that every hourglass-free claw-free graph G of δ(G) ≥ 3 with the single 7-cycle property is Hamiltonian and is best possible; we also show that every claw-free graph G of δ(G) ≥ 3 with the hourglass property and with single 6-cycle property is Hamiltonian.

A note on powers of Hamilton cycles in generalized claw-free graphs

Seymour conjectured for a fixed integer k≥2 that if G is a graph of order n with δ(G)≥kn/(k+1), then G contains the kth power Cnk of a Hamiltonian cycle Cn of G, and this minimum degree condition is sharp. Earlier the k=2 case was conjectured by Pósa. This was verified by Komlós et al. [4]. For s≥3, a graph is K1,s-free if it does not contain an induced subgraph isomorphic to K1,s. Such graphs will be referred to as generalized claw-free graphs. Minimum degree conditions that imply that a generalized claw-free graph G of sufficiently large order n contains a kth power of a Hamiltonian cycle will be proved. More specifically, it will be shown that for any ϵ>0 and for n sufficiently large, any K1,s-free graph of order n with δ(G)≥(1/2+ϵ)n contains a Cnk.

Forbidden Subgraphs for Hamiltonicity of 3-Connected Claw-Free Graphs

In this article, we consider forbidden subgraphs for hamiltonicity of 3-connected claw-free graphs. Let be the graph obtained from a triangle by attaching a path of length i to one of its vertices, and let be the graph obtained from the Petersen graph by adding one pendant edge to each vertex. Lai et al. (J Graph Theory 64(1) (2010), 1–11) conjectured that every 3-connected -free graph G is hamiltonian unless G is the line graph of . It is shown in this article that this conjecture is true. Moreover, we investigate the set of connected graphs which satisfies that every 3-connected -free graph of sufficiently large order is hamiltonian if and only if A is a member of . We prove that, if , then G is a graph on at most 12 vertices with the following structure: (i) a path of length at most 10, (ii) a triangle with three vertex-disjoint paths of total length at most 9, or (iii) G consists of two triangles connected by a path of length 1, 3, 5, or 7. AMS classification: 05C45, 05C38, 05C75. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 146–160, 2013

A simpler proof for vertex-pancyclicity of squares of connected claw-free graphs

We give a simpler proof of the well-known result of Matthews and Sumner stating that squares of connected claw-free graphs are vertex-pancyclic. Contrary to the previous proof, our approach does not resort to Fleischner’s result stating that, when restricted to squares of graphs, vertex-pancyclicity and Hamiltonicity are equivalent. The same proof idea already yielded that connected claw-free graphs of even order have a perfect matching, which is another result of Sumner. We conclude by observing that this proof identifies a larger collection of graphs for which the two properties in question hold.

Collapsible graphs and Hamiltonian connectedness of line graphs

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai [Z.-H. Chen, H.-J. Lai, Reduction techniques for super-Eulerian graphs and related topics—an update, in: Ku Tung-Hsin (Ed.), Combinatorics and Graph Theory, vol. 95, World Scientific, Singapore/London, 1995, pp. 53–69, Conjecture 8.6] conjectured that every 3-edge connected, essentially 6-edge connected graph is collapsible. In this paper, we prove the following results. (1) Every 3-edge connected, essentially 6-edge connected graph with edge-degree at least 7 is collapsible. (2) Every 3-edge connected, essentially 5-edge connected graph with edge-degree at least 6 and at most 24 vertices of degree 3 is collapsible which implies that 5-connected line graph with minimum degree at least 6 of a graph with at most 24 vertices of degree 3 is Hamiltonian. (3) Every 3-connected, essentially 11-connected line graph is Hamilton-connected which strengthens the result in [H.-J. Lai, Y. Shao, H. Wu, J. Zhou, Every 3-connected, essentially 11-connected line graph is Hamiltonian, J. Combin. Theory, Ser. B 96 (2006) 571–576] by Lai et al. (4) Every 7-connected line graph is Hamiltonian connected which is proved by a method different from Zhan’s. By using the multigraph closure introduced by Ryjáček and Vrána which turns a claw-free graph into the line graph of a multigraph while preserving its Hamilton-connectedness, the results (3) and (4) can be extended to claw-free graphs.

The signed maximum-clique transversal number of regular graphs

A function f: V→<texlscub>−1,+1</texlscub>, defined on the vertices of a graph G, is a signed maximum-clique transversal function if ∑ u∈V(Q) f(u)≥1 for each maximum clique Q of G. The weight of a signed maximum-clique transversal function is w(f)=∑ v∈V(G) f(v). The signed maximum-clique transversal number of G, denoted by , is the minimum weight of a signed maximum-clique transversal function of G. In this paper, we establish a tight lower bound on for an arbitrary graph G and an upper bound on for a k-regular graph G with ω(G)=k, and we characterize the extremal graphs achieving the upper bound. Meanwhile, we establish a lower bound on for a connected claw-free cubic graph, and characterize the extremal graphs achieving the lower bound. For a connected claw-free 4-regular graph G with ω(G)=3, we present sharp bounds on and characterize the extremal graphs achieving the lower bound.

Elimination graphs

In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = ( V, E ) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v 1 , …, v n such that the property P holds on the induced subgraph G [ N ( v i )∩ V i ], where N ( v i ) is the neighborhood of v i and V i = { v i , …, v n }. It turns out that if we take P as a graph with maximum independent set size no greater than k , then this definition gives a natural generalization of both chordal graphs and ( k + 1)-claw-free graphs. We refer to such graphs as inductive k -independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k -independent for small k . In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3 -independent graph; furthermore, any planar graph is an inductive 3 -independent graph. For any fixed constant k , we develop simple, polynomial time approximation algorithms for inductive k -independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.

Closure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs

We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and fourth authors, such that the strong closure GM of a claw-free graph G is the line graph of a multigraph containing at most two triangles or at most one double edge.Using the concept of strong closure, we prove that a 3-connected claw-free graph G is Hamilton-connected if G satisfies one of the following: (i) G can be covered by at most 5 cliques, (ii) δ(G)≥4 and G can be covered by at most 6 cliques, (iii) δ(G)≥6 and G can be covered by at most 7 cliques.Finally, by reconsidering the relation between degree conditions and clique coverings in the case of the strong closure GM, we prove that every 3-connected claw-free graph G of minimum degree δ(G)≥24 and minimum degree sum σ8(G)≥n+50 (or, as a corollary, of order n≥142 and minimum degree δ(G)≥n+508) is Hamilton-connected.We also show that our results are asymptotically sharp.

On 3-connected hamiltonian line graphs

Thomassen conjectured that every 4-connected line graph is hamiltonian. It has been proved that every 4-connected line graph of a claw-free graph, or an almost claw-free graph, or a quasi-claw-free graph, is hamiltonian. In 1998, Ainouche et al. [2] introduced the class of DCT graphs, which properly contains both the almost claw-free graphs and the quasi-claw-free graphs. Recently, Broersma and Vumar (2009) [5] found another family of graphs, called P3D graphs, which properly contain all quasi-claw-free graphs. In this paper, we investigate the hamiltonicity of 3-connected line graphs of DCT graphs and P3D graphs, and prove that if G is a DCT graph or a P3D graph with κ(L(G))≥3 and if L(G) does not have an independent vertex 3-cut, then L(G) is hamiltonian. Consequently, every 4-connected line graph of a DCT graph or a P3D graph is hamiltonian.

Minimum Degree and Disjoint Cycles in Claw-Free Graphs

A graph is claw-free if it does not contain an induced subgraph isomorphic to K1,3. Cycles in claw-free graphs have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions. In particular, we prove that if G is claw-free of sufficiently large order n = 3k with δ(G) ≥ n/2, then G contains k disjoint triangles.

On the recognition of fuzzy circular interval graphs

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial time algorithm for recognizing such graphs, and more importantly for building a suitable model for these graphs.