Maria Patricia Dobson

The k-limited packing and k-tuple domination problems in strongly chordal, P4-tidy and split graphs☆

Abstract The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non negative integer k, a subset B of vertices is a k-limited packing if there are at most k elements of B in the closed neighborhood of every vertex. On the other side, a k-tuple domination set in a graph is a subset of vertices D such that every vertex has at least k elements of D in its closed neighborhood. In this work we first reveal a strong relationship between these notions, and obtain from a result due to Liao and Chang (2002), the polynomiality of the k-limited packing problem for strongly chordal graphs. We also prove that, in coincidence with the case of domination, the k-limited packing problem is NP-complete for split graphs. Finally, we prove that both problems are polynomial for the non-perfect class of P4-tidy graphs, including the perfect classes of P4-sparse graphs and cographs.

NP-completeness of the {k}-packing function problem in graphs☆

Abstract Given a positive integer k, the { k } -packing function problem ( { k } PF ) is to find in a given graph G, a function f of maximum weight that assigns a non-negative integer to the vertices of G in such a way that the sum of f ( v ) over each closed neighborhood is at most k. In this work we prove that { k } PF is NP-complete for general graphs. We also expand the set of instances where it is known that { k } PF is linear time solvable, by proving that it is so in dually chordal graphs.

The computational complexity of the Edge-Perfect Graph and the Totally Balanced Packing Game recognition problems

Abstract Edge-perfect graphs were introduced by Escalante et al (2009). An edge-subgraph of a given graph is an induced subgraph obtained by deletion of the endpoints of a subset of edges. A graph is edge-perfect if the stability and the edge covering numbers coincide for every edge-subgraph. In this work we prove that the recognition of edge-perfect graphs is an NP-hard problem. As a by-product, we derive the NP-completeness of two related problems in graphs. From the NP-hardness of the edge-perfection recognition problem we answer the open question on the recognition of totally balanced packing game defining matrices —raised by Deng et al. in 2000—, obtaining that this problem is NP-hard in contrast with the polynomiality for the covering case due to van Velzen (2005).

Treelike Comparability Graphs

Abstract A comparability graph is a simple graph which admits a transitive orientation on its edges. Each one of such orientations defines a poset on the vertex set, and also it is said that this graph is the comparability graph of this poset. A treelike poset is a poset whose covering graph is a tree. Comparability graphs of arborescence posets are known as trivially perfect graphs. These have been characterized by Wolk and Golumbic. In this article we study treelike comparability graphs, that is, comparability graphs of treelike posets. We prove necessary and sufficient conditions that a prime comparability graph must verify for being a treelike comparability graph. Based on the modular decomposition we give a characterization of treelike graphs.

A characterization of edge-perfect graphs and the complexity of recognizing some combinatorial optimization games

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In consequence, recognizing the defining matrices of totally balanced packing games is also co-NP-complete, in contrast with the polynomiality for the covering case. In addition, we solve the computational complexity of universally balanced (with respect to the resources constraints) packing games.

Polynomial reductions between the Limited Packing and Tuple Domination problems in graphs

Abstract The Limited Packing and Tuple Domination problems in graphs have closely-related definitions and the same computational complexity on several graph classes. In this work we present two polynomial reductions between these problems. Thus, by considering graph classes which are closed under these transformations, computational complexity results that are valid for one of the problems give rise to results for the other. The question concerning the existence of a class where one of the problems is polynomial and the other NP-complete is still open.

On transitive orientations with restricted covering graphs

We consider the problem of finding a transitive orientation of a comparability graph, such that the edge set of its covering graph contains a given subset of edges. We propose a solution which employs the classical technique of modular tree decomposition. The method leads to a polynomial time algorithm to construct such an orientation or report that it does not exist.

Towards a Polynomial Equivalence Between \{k\}-Packing Functions and k-Limited Packings in Graphs

Given a positive integer k, the \(\{k\}\)-packing function problem (\(\{k\}\)PF) is to find in a given graph G, a function f of maximum weight that assigns a non-negative integer to the vertices of G in such a way that the sum of f(v) over each closed neighborhood is at most k. This notion was recently introduced as a variation of the k-limited packing problem (kLP) introduced in 2010, where the function was supposed to assign a value in \(\{0,1\}\). For all the graph classes explored up to now, \(\{k\}\)PF and kLP have the same computational complexity. It is an open problem to determine a graph class where one of them is NP-complete and the other, polynomially solvable. In this work, we first prove that \(\{k\}\)PF is NP-complete for bipartite graphs, as kLP is known to be. We also obtain new graph classes where the complexity of these problems would coincide.

Generalized limited packings of some graphs with a limited number of P 4 -partners

By using modular decomposition and handling certain graph operations such as join and union, we show that the Generalized Limited Packing Problem-NP-complete in general-can be solved in polynomial time in some graph classes with a limited number of P 4 -partners; specifically P 4 -tidy graphs, which contain cographs and P 4 -sparse graphs. In particular, we describe an algorithm to compute the associated numbers in polynomial time within these graph classes. In this way, we generalize some of the previous results on the subject. We also make some progress on the study of the computational complexity of the Generalized Multiple Domination Problem in graphs.