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.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

The broadcast median problem in heterogeneous postal model

We propose the problem of finding broadcast medians in heterogeneous networks. A heterogeneous network is represented by a graph G=(V,E), in which each edge has a weight that denotes the communication time between its two end vertices. The overall delay of a vertex v∈V(G), denoted as b(v,G), is the minimum sum of the communication time required to send a message from v to all vertices in G. The broadcast median problem consists of finding the set of vertices v∈V(G) with minimum overall delay b(v,G) and determining the value of b(v,G). In this paper, we consider the broadcast median problem following the heterogeneous postal model. Assuming that the underlying graph G is a general graph, we show that computing b(v,G) for an arbitrary vertex v∈V(G) is NP-hard. On the other hand, assuming that G is a tree, we propose a linear time algorithm for the broadcast median problem in heterogeneous postal model.     

The density maximization problem in graphs

We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G=(V,E) with edge weights w _e ∈? and edge lengths ? _e ∈? for e∈E we define the density of a pattern subgraph H=(V′,E′)?G as the ratio ?(H)=∑ _e∈E′ w _e /∑ _e∈E′ ? _e . We consider the problem of computing a maximum density pattern H under various additional constraints. In doing so, we compute a single Pareto-optimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying bi-objective network construction problem. First, we consider the problem of computing a maximum density pattern with weight at least W and length at most L in a host G. We call this problem the biconstrained density maximization problem. This problem can be interpreted in terms of maximizing the return on investment for network construction problems in the presence of a limited budget and a target profit. We consider this problem for different classes of hosts and patterns. We show that it is NP-hard, even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty. Second, we consider the maximum density subgraph problem under structural constraints on the vertex set that is used by the patterns. While a maximum density perfect matching can be computed efficiently in general graphs, the maximum density Steiner-subgraph problem, which requires a subset of the vertices in any feasible solution, is NP-hard and unlikely to admit a constant-factor approximation. When parameterized by the number of vertices of the pattern, this problem is W[1]-hard in general graphs. On the other hand, it is FPT on planar graphs if there is no constraint on the pattern and on general graphs if the pattern is a path.     

A linear-time algorithm for clique-coloring problem in circular-arc graphs

A maximal clique of G is a clique not properly contained in any other clique. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no maximal clique with at least two vertices is monochromatic. The smallest integer k admitting a k-clique-coloring of G is called clique-coloring number of G. Cerioli and Korenchendler (Electron Notes Discret Math 35:287–292, 2009) showed that there is a polynomial-time algorithm to solve the clique-coloring problem in circular-arc graphs and asked whether there exists a linear-time algorithm to find an optimal clique-coloring in circular-arc graphs or not. In this paper we present a linear-time algorithm of the optimal clique-coloring in circular-arc graphs.     

Vertices in all minimum paired-dominating sets of?block graphs

Let G=(V,E) be a simple graph without isolated vertices. A set S?V is a paired-dominating set if every vertex in V?S has at least one neighbor in S and the subgraph induced by S contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.     

Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S _v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D _v (G,c) the minimum number of integers which must be added to S _v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ _v∈V D _v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.     

An exact algorithm for the maximum probabilistic clique problem

The maximum clique problem is a classical problem in combinatorial optimization that has a broad range of applications in graph-based data mining, social and biological network analysis and a variety of other fields. This article investigates the problem when the edges fail independently with known probabilities. This leads to the maximum probabilistic clique problem, which is to find a subset of vertices of maximum cardinality that forms a clique with probability at least \(\theta \in [0,1]\) , which is a user-specified probability threshold. We show that the probabilistic clique property is hereditary and extend a well-known exact combinatorial algorithm for the maximum clique problem to a sampling-free exact algorithm for the maximum probabilistic clique problem. The performance of the algorithm is benchmarked on a test-bed of DIMACS clique instances and on a randomly generated test-bed.     

Hardness results and approximation algorithm for total liar’s domination in graphs

In this paper, we initiate the study of total liar’s domination of a graph. A subset L?V of a graph G=(V,E) is called a total liar’s dominating set of G if (i) for all v∈V, |N _G (v)∩L|≥2 and (ii) for every pair u,v∈V of distinct vertices, |(N _G (u)∪N _G (v))∩L|≥3. The total liar’s domination number of a graph G is the cardinality of a minimum total liar’s dominating set of G and is denoted by γ _TLR (G). The Minimum Total Liar’s Domination Problem is to find a total liar’s dominating set of minimum cardinality of the input graph G. Given a graph G and a positive integer k, the Total Liar’s Domination Decision Problem is to check whether G has a total liar’s dominating set of cardinality at most k. In this paper, we give a necessary and sufficient condition for the existence of a total liar’s dominating set in a graph. We show that the Total Liar’s Domination Decision Problem is NP-complete for general graphs and is NP-complete even for split graphs and hence for chordal graphs. We also propose a 2(lnΔ(G)+1)-approximation algorithm for the Minimum Total Liar’s Domination Problem, where Δ(G) is the maximum degree of the input graph G. We show that Minimum Total Liar’s Domination Problem cannot be approximated within a factor of $(\frac{1}{8}-\epsilon)\ln(|V|)$ for any ?>0, unless NP?DTIME(|V|^loglog|V|). Finally, we show that Minimum Total Liar’s Domination Problem is APX-complete for graphs with bounded degree 4.     

Profile minimization on compositions of graphs

The profile minimization problem arose from the study of sparse matrix technique. In terms of graphs, the problem is to determine the profile of a graph G which is defined as $$P(G)=\min\limits_{f}\sum\limits_{v\in V(G)}\max\limits_{x\in N[v]}(f(v)-f(x)),$$ where f runs over all bijections from V(G) to {1,2,…,|V(G)|} and N[v]={v}∪{x∈V(G):xv∈E(G)}. This is equivalent to the interval graph completion problem, which is to find a super-graph of a graph G with as few number of edges as possible. The purpose of this paper is to study the profiles of compositions of two graphs.     

Necessary Edges in k-Chordalisations of Graphs

A k-chordalisation of a graph G = (V,E) is a graph H = (V,F) obtained by adding edges to G, such that H is a chordal graph with maximum clique size at most k. This note considers the problem: given a graph G = (V,E) which pairs of vertices, non-adjacent in G, will be an edge in every k-chordalisation of G. Such a pair is called necessary for treewidth k. An equivalent formulation is: which edges can one add to a graph G such that every tree decomposition of G of width at most k is also a tree decomposition of the resulting graph G. Some sufficient, and some necessary and sufficient conditions are given for pairs of vertices to be necessary for treewidth k. For a fixed k, one can find in linear time for a given graph G the set of all necessary pairs for treewidth k. If k is given as part of the input, then this problem is coNP-hard. A few similar results are given when interval graphs (and hence pathwidth) are used instead of chordal graphs and treewidth.     

Some results on the injective chromatic number of?graphs

A k-coloring of a graph G=(V,E) is a mapping c:V??{1,2,??,k}. The coloring c is injective if, for every vertex v??V, all the neighbors of v are assigned with distinct colors. The injective chromatic number ?? _i (G) of G is the smallest k such that G has an injective k-coloring. In this paper, we prove that every K ₄-minor free graph G with maximum degree ????1 has $\chi_{i}(G)\le \lceil \frac{3}{2}\Delta\rceil$ . Moreover, some related results and open problems are given.     

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.     

A continuous characterization of the maximum vertex-weighted clique in hypergraphs

For a simple graph G on n vertices with adjacency matrix A, Motzkin and Strauss established a remarkable connection between the clique number and the global maximum value of the quadratic programm: \(\textit{max}\{ \mathbf {x}^T A \mathbf {x}\}\) on the standard simplex: \(\{\sum _{i=1}^{n} x_i =1, x_i \ge 0 \}\). In Gibbons et al. (Math Oper Res 122:754–768, 1997), an extension of the Motzkin–Straus formulation was provided for the vertex-weighted clique number of a graph. In this paper, we provide a continuous characterization of the maximum vertex-weighted clique problem for vertex-weighted uniform hypergraphs.     

The total {k}-domatic number of wheels and complete graphs

Let k be a positive integer and let G be a graph with vertex set V(G). The total {k}-dominating function (T{k}DF) of a graph G is a function f from V(G) to the set {0,1,2,??,k}, such that for each vertex v??V(G), the sum of the values of all its neighbors assigned by f is at least k. A set {f ₁,f ₂,??,f _d } of pairwise different T{k}DFs of G with the property that $\sum_{i=1}^{d}f_{i}(v)\leq k$ for each v??V(G), is called a total {k}-dominating family (T{k}D family) of G. The total {k}-domatic number of a graph G, denoted by $d_{t}^{\{k\}}(G)$ , is the maximum number of functions in a T{k}D family. In this paper, we determine the exact values of the total {k}-domatic numbers of wheels and complete graphs, which answers an open problem of Sheikholeslami and Volkmann (J. Comb. Optim., 2010) and completes a result in the same paper.     

Max-coloring paths: tight bounds and extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G=(V,E) with vertex weights w such that $\sum_{i=1}^{k}\max_{v\in C_{i}}w(v_{i})$ is minimized, where C ₁,??,C _k are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V|+time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of ??(|V|log?|V|) in the algebraic computation tree model.     

Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs

A k-colouring of a graph G=(V,E) is a mapping c:V→{1,2,…,k} such that c(u)≠c(v) whenever uv is an edge. The reconfiguration graph of the k-colourings of G contains as its vertex set the k-colourings of G, and two colourings are joined by an edge if they differ in colour on just one vertex of G. We introduce a class of k-colourable graphs, which we call k-colour-dense graphs. We show that for each k-colour-dense graph G, the reconfiguration graph of the ?-colourings of G is connected and has diameter O(|V|²), for all ?≥k+1. We show that this graph class contains the k-colourable chordal graphs and that it contains all chordal bipartite graphs when k=2. Moreover, we prove that for each k≥2 there is a k-colourable chordal graph G whose reconfiguration graph of the (k+1)-colourings has diameter Θ(|V|²).     

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F?V, |F|??k, such that G[V?F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 ^O(k) n ^O(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.     

An upper bound on the total restrained domination number of a tree

Let G=(V,E) be a graph. A set of vertices S?V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of $V-\nobreak S$ is adjacent to a vertex in V?S. The total restrained domination number of G, denoted by γ _tr (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that $\gamma_{\mathit{tr}}(T)\leq\lfloor\frac{n+2s+\ell-1}{2}\rfloor$ where T is a tree of order n≥3, and s and ? are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.