María Patricia Dobson - Academia.edu (original) (raw)

Papers by María Patricia Dobson

Theoretical Computer Science, May 1, 2015

Discrete Optimization, Feb 1, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.

International Transactions in Operational Research, Mar 27, 2016

Given a positive integer k, the “‐packing function problem” (PF) is to find in a given graph G, a... more Given a positive integer k, the “‐packing function problem” (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 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 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 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 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.

arXiv (Cornell University), Dec 21, 2018

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex... more The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The k-tuple domination is NP-hard even for chordal graphs. For the class of circulararc graphs, its complexity remains open for k ≥ 2. A 0, 1-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the k-tuple domination problem on graphs G whose augmented adjacency matrix has the C0P for columns, for 2 ≤ k ≤ |U | + 3, where U is the set of universal vertices of G. From an algorithmic point of view, this takes linear time.

Theoretical Computer Science, May 1, 2015

By using modular decomposition and handling certain graph operations such as join and union, we s... more 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 generalcan 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.

Discrete Optimization, Feb 1, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.

International Transactions in Operational Research, Mar 27, 2016

Given a positive integer k, the “‐packing function problem” (PF) is to find in a given graph G, a... more Given a positive integer k, the “‐packing function problem” (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 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 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 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 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.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

arXiv (Cornell University), Dec 21, 2018

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex... more The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The k-tuple domination is NP-hard even for chordal graphs. For the class of circulararc graphs, its complexity remains open for k ≥ 2. A 0, 1-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the k-tuple domination problem on graphs G whose augmented adjacency matrix has the C0P for columns, for 2 ≤ k ≤ |U | + 3, where U is the set of universal vertices of G. From an algorithmic point of view, this takes linear time.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

Information Processing Letters, Dec 1, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Cologne Twente Workshop on Graphs and Combinatorial Optimization, 2009

The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a serie... more The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized biannually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d'Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX,École Polytechnique and CEDRIC, CNAM. As tradition warrants, a special issue of Discrete Applied Mathematics (DAM) will be devoted to CTW09, containing full-length versions of selected presentations given at the workshop and possibly other contributions related to the workshop topics. The deadline for submission to this issue will be posted in due time on the CTW09 website http://www.lix.polytechnique.fr/ctw09.

Information Processing Letters, Dec 1, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

Electronic Notes in Discrete Mathematics, 2015

Abstract Given a positive integer k, the { k } -packing function problem ( { k } PF ) is to find ... more 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.

Information Processing Letters, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Electronic Notes in Discrete Mathematics, 2011

The Limited Packing and Tuple Domination problems in graphs have closely-related definitions and ... more 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

Electronic Notes in Discrete Mathematics, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

Electronic Notes in Discrete Mathematics, 2010

Edge-perfect graphs were introduced by Escalante et al (2009). An edge-subgraph of a given graph ... more 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).

Discrete Optimization, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.

Theoretical Computer Science, May 1, 2015

Discrete Optimization, Feb 1, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.

International Transactions in Operational Research, Mar 27, 2016

Given a positive integer k, the “‐packing function problem” (PF) is to find in a given graph G, a... more Given a positive integer k, the “‐packing function problem” (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 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 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 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 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.

arXiv (Cornell University), Dec 21, 2018

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex... more The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The k-tuple domination is NP-hard even for chordal graphs. For the class of circulararc graphs, its complexity remains open for k ≥ 2. A 0, 1-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the k-tuple domination problem on graphs G whose augmented adjacency matrix has the C0P for columns, for 2 ≤ k ≤ |U | + 3, where U is the set of universal vertices of G. From an algorithmic point of view, this takes linear time.

Theoretical Computer Science, May 1, 2015

By using modular decomposition and handling certain graph operations such as join and union, we s... more 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 generalcan 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.

Discrete Optimization, Feb 1, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.

International Transactions in Operational Research, Mar 27, 2016

Given a positive integer k, the “‐packing function problem” (PF) is to find in a given graph G, a... more Given a positive integer k, the “‐packing function problem” (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 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 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 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 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.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

arXiv (Cornell University), Dec 21, 2018

The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex... more The k-tuple domination problem, for a fixed positive integer k, is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The k-tuple domination is NP-hard even for chordal graphs. For the class of circulararc graphs, its complexity remains open for k ≥ 2. A 0, 1-matrix has the consecutive 0's property (C0P) for columns if there is a permutation of its rows that places the 0's consecutively in every column. Due to A. Tucker, graphs whose augmented adjancency matrix has the C0P for columns are circular-arc. In this work we study the k-tuple domination problem on graphs G whose augmented adjacency matrix has the C0P for columns, for 2 ≤ k ≤ |U | + 3, where U is the set of universal vertices of G. From an algorithmic point of view, this takes linear time.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

Information Processing Letters, Dec 1, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Cologne Twente Workshop on Graphs and Combinatorial Optimization, 2009

The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a serie... more The Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized biannually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d'Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX,École Polytechnique and CEDRIC, CNAM. As tradition warrants, a special issue of Discrete Applied Mathematics (DAM) will be devoted to CTW09, containing full-length versions of selected presentations given at the workshop and possibly other contributions related to the workshop topics. The deadline for submission to this issue will be posted in due time on the CTW09 website http://www.lix.polytechnique.fr/ctw09.

Information Processing Letters, Dec 1, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Electronic Notes in Discrete Mathematics, Aug 1, 2010

Electronic Notes in Discrete Mathematics, 2015

Abstract Given a positive integer k, the { k } -packing function problem ( { k } PF ) is to find ... more 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.

Information Processing Letters, 2011

In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduc... more In this work we confront-from a computational viewpoint-the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P 4-tidy graphs. From this result, we derive that they are polynomial time solvable in P 4-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side.

Electronic Notes in Discrete Mathematics, 2011

The Limited Packing and Tuple Domination problems in graphs have closely-related definitions and ... more 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

Electronic Notes in Discrete Mathematics, 2010

The notion of k-limited packing in a graph is a generalization of 2-packing. For a given non nega... more 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 P 4-tidy graphs, including the perfect classes of P 4-sparse graphs and cographs.

Electronic Notes in Discrete Mathematics, 2010

Edge-perfect graphs were introduced by Escalante et al (2009). An edge-subgraph of a given graph ... more 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).

Discrete Optimization, 2013

We characterize edge-perfect graphs and prove that it is co-NP-complete to recognize them. In con... more 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.