Claw-free Graphs Research Articles

Stabex method for extension of α-polynomial hereditary classes

A class of graphs is called α- polynomial if there exists a polynomial-time algorithm for finding the stability number α( G) for all graphs G in the class. We define a set Stabex( F, v) associated with a graph F and a vertex v∈ V( F). Let F be a minimal forbidden induced subgraph for an α-polynomial hereditary class P . It is shown that if we change F with Stabex( F, v) in the list of forbidden induced subgraphs for P , then we obtain an α-polynomial extension of P . We obtain a generalization of Sbihi’s theorem [Discrete Math. 28(3) (1980) 53] on the stability number problem in claw-free graphs. Also, a recent result of Lozin [Eur. J. Oper. Res. 125 (2000) 292] on stability in ( P 5, P)-free graphs is a corollary of our result.

On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].

On traceability and 2-factors in claw-free graphs

On traceability and 2-factors in claw-free graphs

HamiltonianN2-locally connected claw-free graphs

HamiltonianN2-locally connected claw-free graphs

Independence and upper irredundance in claw-free graphs

It is known that the independence number of a connected claw-free graph G of order n is at most ( n+1)/2. We improve this result by showing that this bound still holds for the upper irredundance number IR( G). We characterize the connected claw-free graphs for which IR( G)=( n+1)/2 and give some properties of those graphs for which IR( G)= n/2 if n is even or IR( G)=( n−1)/2 if n is odd.

Robust Algorithms for the Stable Set Problem

The stable set problem is to find in a simple graph a maximum subset of pairwise non-adjacent vertices. The problem is known to be NP-hard in general and can be solved in polynomial time on some special classes, like cographs or claw-free graphs. Usually, efficient algorithms assume membership of a given graph in a special class. Robust algorithms apply to any graph G and either solve the problem for G or find in it special forbidden configurations. In the present paper we describe several efficient robust algorithms, extending some known results.

Clique family inequalities for the stable set polytope of quasi-line graphs

In one of fundamental works in combinatorial optimization, Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets in its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs are a subclass of claw-free graphs, and as for claw-free graphs, there exists a polynomial algorithm for finding a maximum weighted stable set in such graphs, but we do not have a complete characterization of their stable set polytope (SSP). In the paper, we introduce a class of inequalities, called clique-family inequalities, which are valid for the SSP of any graph and match the odd set inequalities defined by Edmonds for the matching polytope. This class of inequalities unifies all the known (non-trivial) facet inducing inequalities for the SSP of a quasi-line graph. We, therefore, conjecture that all the non-trivial facets of the SSP of a quasi-line graph belong to this class. We show that the conjecture is indeed correct for the subclasses of quasi-line graphs for which we have a complete description of the SSP. We discuss some approaches for solving the conjecture and a related problem.

On graph closures

Ryjáček (J. Combin. Theory B 70 (1997) 217) introduced a very useful notion of a closure cl( G) for a claw-free graph G and proved, in particular, that c( G)= c(cl( G)) where c( H) is the length of a longest cycle in H. In this paper, we describe some strengthenings of the main results in Ryjáček (J. Combin. Theory B 70 (1997) 217). As a result, we introduce some new closures σ( G) that can be used in a wider class of graphs, and show, in particular, that c( G)= c( σ( G)) and p( G)= p( σ( G)) where p( H) is the length of a longest path in H. As a byproduct, we give some new sufficient conditions for graphs to have a Hamiltonian cycle, path, v-path, and uv-path, and show, in particular, that every claw-free 9-connected graph is Hamiltonian connected. We also give a construction that provides infinitely many counterexamples to the conjecture on the so-called cl 2-closure in Bollobas et al. (Discrete Math. 195 (1999) 67).

Interval degree and bandwidth of a graph

The interval degree id( G) of a graph G is defined to be the smallest max-degree of any interval supergraphs of G. One of the reasons for our interest in this parameter is that the bandwidth of a graph is always between id( G)/2 and id( G). We prove also that for any graph G the interval degree of G is at least the pathwidth of G 2. We show that if G is an AT-free claw-free graph, then the interval degree of G is equal to the clique number of G 2 minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs.

Finding Maximum Induced Matchings in Subclasses of Claw-Free and P 5-Free Graphs, and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P5,Dm)-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.

Triangle-factors in powers of graphs

Abstract In this paper, we investigate existence of triangle-factors, that is 2-factors in which every cycle is of length 3, in powers of graphs of order 3 k , k ≥ 1. It is easy to show that G 4 contains a triangle-factor for any connected graph and G 3 contains a triangle-factor for any connected claw-free graph G Our main result is that G 3 contains a triangle-factor for any 2-connected graph G. We also show, using the same method, that any 2-connected claw-free graph contains a P 3 -factor which implies that the square of the graph contains a triangle-factor.

Finding augmenting chains in extensions of claw-free graphs

Finding augmenting chains is in the heart of the maximum matching problem, which is equivalent to the maximum stable set problem in the class of line graphs. Due to the celebrated result of Edmonds, augmenting chains can be found in line graphs in polynomial time. Minty and Sbihi generalized this result to claw-free graphs. In this paper we extend it to larger classes. As a particular consequence, a new polynomially solvable case for the maximum stable set problem has been detected.

On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four–Extended Abstract–

Abstract Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ≤ 3 have rank facets only [3,12] while there are examples with clique number > 4 having non-rank facets [8,10]. The aim of the present paper is to treat the remaining case with clique number = 4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets.

Every 3-connected distance claw-free graph is Hamilton-connected

A graph G is said to be distance claw-free (in DC) if for each vertex v in V( G) the independence number of the subgraph of G induced by the set of vertices at distance i from v is at most 2 for each i⩾0. It is shown that every 3-connected graph in DC is hamilton-connected, answering affirmatively a question of Shepherd.

Traceability in Small Claw-Free Graphs

Abstract We prove that a claw-free, 2-connected graph with fewer than 18 vertices is traceable, and we determine all non-traceable, claw-free, 2-connected graphs with exactly 18 vertices and a minimal number of edges. This complements a result of Matthews on Hamiltonian graphs.

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

Minty has shown that the Maximum Weight Stable Set (MWS) Problem can be solved in polynomial time when restricted to claw-free graphs. We show that the structure of graphs being both claw-free and co-claw-free is very simple which implies bounded clique width for this graph class. It is known that for graph classes of bounded clique width (assuming that k-expressions can be determined in linear time), there are linear time algorithms for all problems expressible in Monadic Second Order Logic with quantification only over vertex sets. The problems MWS, Maximum Clique, Domination and Steiner Tree, for example, are expressible in this way. Moreover, we describe the structure of prime graphs being both H- and H -free for any four-vertex graph H and obtain bounded clique width for these graph classes except the 2K 2 - and C 4 -free graphs.

On some intriguing problems in hamiltonian graph theory—a survey

We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.

Hamiltonian cycles in 3-connected Claw-free graphs

It is shown that every 3-connected claw-free graph having at most 6 δ−7 vertices is hamiltonian, where δ is the minimum degree.

A note on degree conditions for hamiltonicity in 2-connected claw-free graphs

Let G be a claw-free graph and let cl(G) be the closure of G. We present a method for characterizing classes G i , i=3,…,7, of 2-connected closed claw-free graphs with the following properties. (i) Theorem. Let G be a 2- connected claw-free graph of order n⩾153 such that δ(G)⩾20 and σ 8(G)>n+39 . Then either G is hamiltonian or cl (G)∈⋃ i=3 7 G i . (ii) Corollary. Let G be a 2- connected claw-free graph of order n⩾153 with δ(G)⩾(n+39)/8. Then either G is hamiltonian or cl (G)∈⋃ i=3 7 G i . The family of exceptions contains 318 infinite classes. The majority of these exception classes were found with the help of a computer.

Pancyclicity in claw-free graphs

In this paper, we present several conditions for K 1,3-free graphs, which guarantee the graph is subpancyclic. In particular, we show that every K 1,3-free graph with a minimum degree sum δ 2>2 3n+1 −4 ; every { K 1,3, P 7}-free graph with δ 2⩾9; every { K 1,3, Z 4}-free graph with δ 2⩾9; and every K 1,3-free graph with maximum degree Δ, diam( G)<( Δ+6)/4 and δ 2⩾9 is subpancyclic.