Valeria A. Leoni

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.

On the complexity of { k } -domination and k-tuple domination in graphs

We consider two types of graph domination- { k } -domination and k-tuple domination, for a fixed positive integer k-and provide new NP-complete as well as polynomial time solvable instances for their related decision problems. Regarding NP-completeness results, we solve the complexity of the { k } -domination problem on split graphs, chordal bipartite graphs and planar graphs, left open in 2008. On the other hand, by exploiting Courcelles results on Monadic Second Order Logic, we obtain that both problems are polynomial time solvable for graphs with clique-width bounded by a constant. In addition, we give an alternative proof for the linearity of these problems on strongly chordal graphs.

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.

On the complexity of the {k}‐packing function problem

Given a positive integer k, the “{k}-packing function problem” ({k}PF) is to find in a given graph G, a function f that assigns a nonnegative integer to the vertices of G in such a way that the sum of f (v) over each closed neighborhood is at most k and over the whole vertex set of G (weight of f ) is maximum. It is known that {k}PF is linear time solvable in strongly chordal graphs and in graphs with clique-width bounded by a constant. In this paper we prove that {k}PF is NP-complete, even when restricted to chordal graphs that constitute a superclass of strongly chordal graphs. To find other subclasses of chordal graphs where {k}PF is tractable, we prove that it is linear time solvable for doubly chordal graphs, by proving that it is so in the superclass of dually chordal graphs, which are graphs that have a maximum neighborhood ordering.

Limited Packing and Multiple Domination problems: Polynomial time reductions

The Limited Packing and Multiple Domination problems in graphs have closely-related definitions and the same computational complexity on several graph classes. In this work we present two polynomial time reductions between them. Besides, we take into consideration generalized versions of these problems and obtain polynomial time reductions between each one and its generalized version.

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).

\(\{k\}\)-Packing Functions of Graphs

Given a positive integer \(k\) and a graph \(G\), a \(k\)-limited packing in \(G\) (2010) is a subset \(B\) of its vertex set such that each closed neighborhood has at most \(k\) vertices of \(B\). As a variation, we introduce the notion of a \(\{k\}\)-packing function \(f\) of \(G\) which 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 \(k\). For fixed \(k\), we prove that the problem of finding a \(\{k\}\)-packing function of maximum weight (\(\{k\)}PF) can be reduced linearly to the problem of finding a \(k\)-limited packing of maximum cardinality (\(k\)LP). We present an \(O(|V(G)|+|E(G)|)\) time algorithm to solve \(\{k\)}PF on strongly chordal graphs. We also use monadic second-order logic to prove that both problems are linear time solvable for graphs with clique-width bounded by a constant.

{k}-domination for chordal graphs and related graph classes☆

Abstract In this work we obtain a new graph class where the { k } -dominating function problem ( { k } -DOM) is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. Firstly, by relating this problem with the k-dominating function problem (k-DOM), we prove that { k } -DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second-order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. Finally, we show that { k } -DOM is linear time solvable for spider 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.