The densest <Emphasis Type="Italic">k</Emphasis>-subgraph problem on clique graphs

The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The problem is strongly NP-hard, as a generalization of the well known Clique problem and we also know that it does not admit a Polynomial Time Approximation Scheme (PTAS). In this paper we focus on special cases of the problem, with respect to the class of the input graph. Especially, towards the elucidation of the open questions concerning the complexity of the problem for interval graphs as well as its approximability for chordal graphs, we consider graphs having special clique graphs. We present a PTAS for stars of cliques and a dynamic programming algorithm for trees of cliques. M.L. is co-financed within Op. Education by the ESF (European Social Fund) and National Resources. V.Z. is partially supported by the Special Research Grants Account of the University of Athens under Grant 70/4/5821.     

Labelling algorithms for paired-domination problems in block and interval graphs

Let G=(V,E) be a graph without isolated vertices. A set S⊆V is a paired-dominating set if every vertex in V−S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng (Discrete Appl. Math. 155:2077–2086, 2007), we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is NP-complete for bipartite graphs, chordal graphs, even for split graphs.     

Minimum paired-dominating set in chordal bipartite graphs and perfect elimination bipartite graphs

A set D?V of a graph G=(V,E) is a dominating set of G if every vertex in V?D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if the induced subgraph, G[D], has a perfect matching. Given a graph G=(V,E) and a positive integer k, the paired-domination problem is to decide whether G has a paired-dominating set of cardinality at most k. The paired-domination problem is known to be NP-complete for bipartite graphs. In this paper, we, first, strengthen this complexity result by showing that the paired-domination problem is NP-complete for perfect elimination bipartite graphs. We, then, propose a linear time algorithm to compute a minimum paired-dominating set of a chordal bipartite graph, a well studied subclass of bipartite graphs.     

The max quasi-independent set problem

In this paper, we deal with the problem of finding quasi-independent sets in graphs. This problem is formally defined in three versions, which are shown to be polynomially equivalent. The one that looks most general, namely, f-max quasi-independent set, consists of, given a graph and a non-decreasing function f, finding a maximum size subset Q of the vertices of the graph, such that the number of edges in the induced subgraph is less than or equal to f(|Q|). For this problem, we show an exact solution method that runs within time O^*(2^{\fracd-27/23d+1n})O^{*}(2^{\frac{d-27/23}{d+1}n}) on graphs of average degree bounded by d. For the most specifically defined γ-max quasi-independent set and k-max quasi-independent set problems, several results on complexity and approximation are shown, and greedy algorithms are proposed, analyzed and tested.     

Maximum <Emphasis Type="Italic">k</Emphasis>-regular induced subgraphs

Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any fixed value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we extend the Hoffman bound on the size of a maximum 0-regular subgraph (the independence number) from k=0 to larger values of k.     

Compact labelings for efficient first-order model-checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is nicely locally clique-width-decomposable. This notion generalizes that of a nicely locally tree-decomposable class. The graphs of such classes can be covered by graphs of bounded clique-width with limited overlaps. We also consider such labelings for bounded first-order formulas on graph classes of bounded expansion. Some of these results are extended to counting queries.     

On the distance paired domination of generalized Petersen graphs <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,1) and <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,2)

Let G=(V,E) be a graph without an isolated vertex. A set D⊆V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by γ _p ^k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k≥1.     

Exact combinatorial algorithms and experiments for?finding maximum k-plexes

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.     

Penta-Extensions of Hereditary Classes of Graphs

Penta is the configuration shown in figure 1(a), where continuous lines represent edges and dotted lines represent non-edges. The vertex u in figure 1(a) is called the center of Penta. A graph G is called a pentagraph if every induced subgraph H of G has a vertex v which is not a center of induced Penta in H. The class of pentagraphs is a common generalization of chordal [triangulated] graphs and Mahadev graphs. We construct a polynomial-time algorithm that either find a maximum stable set of G or concludes that G is not a pentagraph. We propose a method for extending α-polynomial hereditary classes based on induced Pentas.     

Structure of Fibonacci cubes: a survey

The Fibonacci cube Γ _n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.     

The <Emphasis Type="Italic">k</Emphasis>-hop connected dominating set problem: approximation and hardness

Let G be a connected graph and k be a positive integer. A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G there is a vertex u in D such that the distance between v and u in G is at most k. We study the problem of finding a minimum k-hop connected dominating set of a graph (\({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\)). We prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {NP}\)-hard on planar bipartite graphs of maximum degree 4. We also prove that \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) is \(\mathscr {APX}\)-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\) on bipartite and on (1, 2)-split graphs. Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices. We also discuss the complexity of computing this graph parameter. On the positive side, we show an approximation algorithm for \({\textsc {Min}}k{\hbox {-}\textsc {CDS}}\). Finally, when \(k=1\), we present two new approximation algorithms for the weighted version of the problem restricted to graphs with a polynomially bounded number of minimal separators.     

Algorithm complexity of neighborhood total domination and $$(\rho ,\gamma _{nt})$$-graphs

A neighborhood total dominating set, abbreviated for NTD-set D, is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by \(\gamma _{nt}(G)\), is the minimum cardinality of a NTD-set in G. In this paper, we prove that NTD problem is NP-complete for bipartite graphs and split graphs. Then we give a linear-time algorithm to determine \(\gamma _{nt}(T)\) for a given tree T. Finally, we characterize a constructive property of \((\gamma _{nt},2\gamma )\)-trees and provide a constructive characterization for \((\rho ,\gamma _{nt})\)-graphs, where \(\gamma \) and \(\rho \) are domination number and packing number for the given graph, respectively.     

An adaptive multistart tabu search approach to solve the maximum clique problem

Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.     

The optimum traversal of a graph

For a given graph (network) having costs [c_ij] associated with its links, the present paper examines the problem of finding a cycle which traverses every link of the graph at least once, and which incurs the minimum cost of traversal. This problem (called thegraph traversal problem, or theChinese postman problem [9]) can be formulated in ways analogous to those used for the well-known travelling salesman problem, and using this apparent similarity, Bellman and Cooke [1] have produced a dynamic programming formulation. This method of solution of the graph traversal problem requires computational times which increase exponentially with the number of links in the graph. Approximately the same rate of increase of computational effort with problem size would result by any other method adapting a travelling salesman algorithm to the present problem.This paper describes an efficient algorithm for the optimal solution of the graph traversal problem based on the matching method of Edmonds [5, 6]. The computational time requirements of this algorithm increase as a low order (2 or 3) power of the number of links in the graph. Computational results are given for graphs of up to 50 vertices and 125 links.The paper then discusses a generalised version of the graph traversal problem, where not one but a number of cycles are required to traverse the graph. In this case each link has (in addition to its cost) a quantity q_ij associated with it, and the sum of the quantities of the links in any one cycle must be less than a given amount representing the cycle capacity. A heuristic algorithm for the solution of this problem is given. The algorithm is based on the optimal algorithm for the single-cycle graph traversal problem and is shown to produce near-optimal results.There is a large number of possible applications where graph traversal problems arise. These applications include: the spraying of roads with salt-grit to prevent ice formation, the inspection of electric power lines, gas, or oil pipelines for faults, the delivery of letter post, etc.     

Half integer extreme points in the linear relaxation of the 2-edge-connected subgraph polyhedron

This paper studies the graphs for which the linear relaxation of the 2-connected spanning subgraph polyhedron has integer or half-integer extreme points. These graphs are called quasi-integer. For these graphs, the linear relaxation of the k-edge connected spanning subgraph polyhedron is integer for all k=4r, r≥1. The class of quasi-integer graphs is closed under minors and contains for instance the class of series-parallel graphs. We discuss some structural properties of graphs which are minimally non quasi-integer graphs, then we examine some basic operations which preserve the quasi-integer property. Using this, we show that the subdivisions of wheels are quasi-integer.     

Composition of Graphs and the Triangle-Free Subgraph Polytope

In this paper, we study a composition (decomposition) technique for the triangle-free subgraph polytope in graphs which are decomposable by means of 3-sums satisfying some property. If a graph G decomposes into two graphs G ₁ and G ₂, we show that the triangle-free subgraph polytope of G can be described from two linear systems related to G ₁ and G ₂. This gives a way to characterize this polytope on graphs that can be recursively decomposed. This also gives a procedure to derive new facets for this polytope. We also show that, if the systems associated with G ₁ and G ₂ are TDI, then the system characterizing the polytope for G is TDI. This generalizes previous results in R. Euler and A.R. Mahjoub (Journal of Comb. Theory series B, vol. 53, no. 2, pp. 235–259, 1991) and A.R. Mahjoub (Discrete Applied Math., vol. 62, pp. 209–219, 1995).     

On risk-averse maximum weighted subgraph problems

In this work, we consider a class of risk-averse maximum weighted subgraph problems (R-MWSP). Namely, assuming that each vertex of the graph is associated with a stochastic weight, such that the joint distribution is known, the goal is to obtain a subgraph of minimum risk satisfying a given hereditary property. We employ a stochastic programming framework that is based on the formalism of modern theory of risk measures in order to find minimum-risk hereditary structures in graphs with stochastic vertex weights. The introduced form of risk function for measuring the risk of subgraphs ensures that optimal solutions of R-MWS problems represent maximal subgraphs. A graph-based branch-and-bound (BnB) algorithm for solving the proposed problems is developed and illustrated on a special case of risk-averse maximum weighted clique problem. Numerical experiments on randomly generated Erdös-Rényi graphs demonstrate the computational performance of the developed BnB.     

Algorithms for the maximum k-club problem in graphs

Given a simple undirected graph G, a k-club is a subset of vertices inducing a subgraph of diameter at most k. The maximum k-club problem (MkCP) is to find a k-club of maximum cardinality in G. These structures, originally introduced to model cohesive subgroups in social network analysis, are of interest in network-based data mining and clustering applications. The maximum k-club problem is NP-hard, moreover, determining whether a given k-club is maximal (by inclusion) is NP-hard as well. This paper first provides a sufficient condition for testing maximality of a given k-club. Then it proceeds to develop a variable neighborhood search (VNS) heuristic and an exact algorithm for MkCP that uses the VNS solution as a lower bound. Computational experiments with test instances available in the literature show that the proposed algorithms are very effective on sparse instances and outperform the existing methods on most dense graphs from the testbed.     

Vertex and Tree Arboricities of Graphs

This paper studies the following variations of arboricity of graphs. The vertex (respectively, tree) arboricity of a graph G is the minimum number va(G) (respectively, ta(G)) of subsets into which the vertices of G can be partitioned so that each subset induces a forest (respectively, tree). This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness. The graphs investigated in this paper include block-cactus graphs, series-parallel graphs, cographs and planar graphs.     

A simplex like approach based on star sets for recognizing convex-QP adverse graphs

A graph \(G\) with convex-\(QP\) stability number (or simply a convex-\(QP\) graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say \(P(G)\). There are polynomial-time procedures to recognize convex-\(QP\) graphs, except when the graph \(G\) is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the least eigenvalue of its adjacency matrix and the optimal value of \(P(G)\) are both integer and none of them changes when the neighborhood of any vertex of \(G\) is deleted). In this paper, from a characterization of convex-\(QP\) graphs based on star sets associated to the least eigenvalue of its adjacency matrix, a simplex-like algorithm for the recognition of convex-\(QP\) adverse graphs is introduced.