Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. Recently we were able to characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. These characterizations lead to polynomial time recognition of clique-perfect graphs in these classes of graphs. In this paper we solve the characterization problem in two new classes of graphs: diamond-free and Helly circular-arc () graphs. This last characterization leads to a polynomial time recognition algorithm for clique-perfect graphs.     

Self-clique Helly circular-arc graphs

A clique in a graph is a complete subgraph maximal under inclusion. The clique graph of a graph is the intersection graph of its cliques. A graph is self-clique when it is isomorphic to its clique graph. A circular-arc graph is the intersection graph of a family of arcs of a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. In this note, we describe all the self-clique Helly circular-arc graphs.     

Solving the path cover problem on circular-arc graphs by using an approximation algorithm

A path cover of a graph G=(V,E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs.     

The clique operator on circular-arc graphs

A circular-arc graphG is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graphK(G) of a graph G is the intersection graph of its cliques. The iterated clique graphKⁱ(G) of G is defined by K⁰(G)=G and Kⁱ⁺¹(G)=K(Kⁱ(G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K-converges, if it is K-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.     

The exact domination number of the generalized Petersen graphs

Let G=(V,E) be a graph. A subset S⊆V is a dominating set of G, if every vertex u∈V−S is dominated by some vertex v∈S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2008) 603-610] proved that and conjectured that the upper bound is the exact domination number. In this paper we prove this conjecture.     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

Recognizing and representing proper interval graphs in parallel using merging and sorting

We present a parallel algorithm for recognizing and representing a proper interval graph in time with O(m+n) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from which an interval representation is easily obtained. It is simple, uses no complex data structures, and extends ideas from an optimal sequential algorithm for recognizing and representing a proper interval graph [X. Deng, P. Hell, J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput. 25 (2) (1996) 390-403].     

Improved induced matchings in sparse graphs

An induced matching in a graph G=(V,E) is a matching M such that (V,M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al. [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least and that there are twinless planar graphs that do not contain an induced matching of size greater than . We improve both these bounds to , which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity.Kanj et al. also presented an algorithm which decides in -time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient -time algorithm. Its main ingredient is a new, O^∗(4^l)-time algorithm for finding a maximum induced matching in a graph of branch width at most l.     

Zero-divisor semigroups and refinements of a star graph

Let G be a refinement of a star graph with center c. Let be the subgraph of G induced on the vertex set V(G)?{c or end vertices adjacent to c}. In this paper, we completely determine the structure of commutative zero-divisor semigroups S whose zero-divisor graph G=Γ(S) and S satisfy one of the following properties: (1) has at least two connected components, (2) is a cycle graph C_n of length n≥5, (3) is a path graph P_n with n≥6, (4) S is nilpotent and Γ(S) is a finite or an infinite star graph. For any finite or infinite cardinal number n≥2, we prove that for any nilpotent semigroup S with zero element 0, S⁴=0 if Γ(S) is a star graph K_1,n. We prove that there is exactly one nilpotent semigroup S such that S³≠0 and Γ(S)≅K_1,n. For several classes of finite graphs G which are refinements of a star graph, we also obtain formulas to count the number of non-isomorphic corresponding semigroups.     

An upper bound on the paired-domination number in terms of the number of edges in the graph

In this paper, we continue the study of paired domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998) 199-206]. A paired-dominating set of a graph is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by , is the minimum cardinality of a paired-dominating set in G. We show that if G is a connected graph of size m≥18 with minimum degree at least 2, then and we characterize the (infinite family of) graphs that achieve equality in this bound.     

On the parallel computation of the biconnected and strongly connected co-components of graphs

In this paper, we consider the problems of co-biconnectivity and strong co-connectivity, i.e., computing the biconnected components and the strongly connected components of the complement of a given graph. We describe simple sequential algorithms for these problems, which work on the input graph and not on its complement, and which for a graph on n vertices and m edges both run in optimal O(n+m) time. Our algorithms are not data structure-based and they employ neither breadth-first-search nor depth-first-search.Unlike previous linear co-biconnectivity and strong co-connectivity sequential algorithms, both algorithms admit efficient parallelization. The co-biconnectivity algorithm can be parallelized resulting in an optimal parallel algorithm that runs in time using processors. The strong co-connectivity algorithm can also be parallelized to yield an -time and O(m^1.188/logn)-processor solution. As a byproduct, we obtain a simple optimal O(logn)-time parallel co-connectivity algorithm.Our results show that, in a parallel process environment, the problems of computing the biconnected components and the strongly connected components can be solved with better time-processor complexity on the complement of a graph rather than on the graph itself.     

On the diameter of generalized Kneser graphs

Let r, k be positive integers, s(<r), a nonnegative integer, and n=2r-s+k. The set of r-subsets of [n]={1,2,…,n} is denoted by [n]^r. The generalized Kneser graph K(n,r,s) is the graph whose vertex-set is [n]^r where two r-subsets A and B are joined by an edge if |A∩B|?s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n,r,s) is equal to , which generalizes a result of Valencia-Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

Improved bottleneck domination algorithms

W.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an -time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs.We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression.     

A note on the chromatic number of a dense random graph

Let G_n,p denote the random graph on n labeled vertices, where each edge is included with probability p independent of the others. We show that for all constant p

     

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K₄, then . In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then unless G is isomorphic to K₄ (in which case its boxicity is 1).     

Integral point sets in higher dimensional affine spaces over finite fields

We consider point sets in the m-dimensional affine space where each squared Euclidean distance of two points is a square in Fq. It turns out that the situation in is rather similar to the one of integral distances in Euclidean spaces. Therefore we expect the results over finite fields to be useful for the Euclidean case.We completely determine the automorphism group of these spaces which preserves integral distances. For some small parameters m and q we determine the maximum cardinality I(m,q) of integral point sets in . We provide upper bounds and lower bounds on I(m,q). If we map integral distances to edges in a graph, we can define a graph Gm,q with vertex set . It turns out that Gm,q is strongly regular for some cases.     

On orthogonal ray graphs

An orthogonal ray graph is an intersection graph of horizontal and vertical rays (half-lines) in the xy-plane. An orthogonal ray graph is a 2-directional orthogonal ray graph if all the horizontal rays extend in the positive x-direction and all the vertical rays extend in the positive y-direction. We first show that the class of orthogonal ray graphs is a proper subset of the class of unit grid intersection graphs. We next provide several characterizations of 2-directional orthogonal ray graphs. Our first characterization is based on forbidden submatrices. A characterization in terms of a vertex ordering follows immediately. Next, we show that 2-directional orthogonal ray graphs are exactly those bipartite graphs whose complements are circular arc graphs. This characterization implies polynomial-time recognition and isomorphism algorithms for 2-directional orthogonal ray graphs. It also leads to a characterization of 2-directional orthogonal ray graphs by a list of forbidden induced subgraphs. We also show a characterization of 2-directional orthogonal ray trees, which implies a linear-time algorithm to recognize such trees. Our results settle an open question of deciding whether a (0,1)-matrix can be permuted to avoid the submatrices .     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

Sequences with small subsum sets

A conjecture of Gao and Leader, recently proved by Sun, states that if is a sequence of length n in a finite abelian group of exponent n, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 2n−1. This conjecture turns out to be a simple consequence of a theorem of Olson and White; we investigate generalizations that are not implied by this theorem. In particular, we prove the following result: if is a sequence of length n, the terms of which generate a finite abelian group of rank at least 3, then either some subsequence of X sums to zero or the set of all sums of subsequences of X has cardinality at least 4n−5.