Displaying 21 – 40 of 51

Showing per page

A note on domination parameters in random graphs

Anthony Bonato, Changping Wang (2008)

Discussiones Mathematicae Graph Theory

Domination parameters in random graphs G(n,p), where p is a fixed real number in (0,1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n,p).

A note on domination parameters of the conjunction of two special graphs

Maciej Zwierzchowski (2001)

Discussiones Mathematicae Graph Theory

A dominating set D of G is called a split dominating set of G if the subgraph induced by the subset V(G)-D is disconnected. The cardinality of a minimum split dominating set is called the minimum split domination number of G. Such subset and such number was introduced in [4]. In [2], [3] the authors estimated the domination number of products of graphs. More precisely, they were study products of paths. Inspired by those results we give another estimation of the domination number of the conjunction...

A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation

Marcia R. Cerioli, Luerbio Faria, Talita O. Ferreira, Fábio Protti (2011)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

A unit disk graph is the intersection graph of a family of unit disks in the plane. If the disks do not overlap, it is also a unit coin graph or penny graph. It is known that finding a maximum independent set in a unit disk graph is a NP-hard problem. In this work we extend this result to penny graphs. Furthermore, we prove that finding a minimum clique partition in a penny graph is also NP-hard, and present two linear-time approximation algorithms for the computation of clique partitions: a 3-approximation...

A note on maximum independent sets and minimum clique partitions in unit disk graphs and penny graphs: complexity and approximation

Marcia R. Cerioli, Luerbio Faria, Talita O. Ferreira, Fábio Protti (2011)

RAIRO - Theoretical Informatics and Applications

A unit disk graph is the intersection graph of a family of unit disks in the plane. If the disks do not overlap, it is also a unit coin graph or penny graph. It is known that finding a maximum independent set in a unit disk graph is a NP-hard problem. In this work we extend this result to penny graphs. Furthermore, we prove that finding a minimum clique partition in a penny graph is also NP-hard, and present two linear-time approximation algorithms for the computation of clique partitions: a 3-approximation...

A Note on Non-Dominating Set Partitions in Graphs

Wyatt J. Desormeaux, Teresa W. Haynes, Michael A. Henning (2016)

Discussiones Mathematicae Graph Theory

A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is...

A Note on Path Domination

Liliana Alcón (2016)

Discussiones Mathematicae Graph Theory

We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.

A note on the Chvátal-rank of clique family inequalities

Arnaud Pêcher, Annegret K. Wagler (2007)

RAIRO - Operations Research


Clique family inequalities a∑v∈W xv + (a - 1)∈v∈W, xv ≤ aδ form an intriguing class of valid inequalities for the stable set polytopes of all graphs. We prove firstly that their Chvátal-rank is at most a, which provides an alternative proof for the validity of clique family inequalities, involving only standard rounding arguments. Secondly, we strengthen the upper bound further and discuss consequences regarding the Chvátal-rank of subclasses of claw-free graphs.


A note on the domination number of a graph and its complement

Dănuţ Marcu (2001)

Mathematica Bohemica

If G is a simple graph of size n without isolated vertices and G ¯ is its complement, we show that the domination numbers of G and G ¯ satisfy γ ( G ) + γ ( G ¯ ) n - δ + 2 if γ ( G ) > 3 , δ + 3 if γ ( G ¯ ) > 3 , where δ is the minimum degree of vertices in G .

A note on the double Roman domination number of graphs

Xue-Gang Chen (2020)

Czechoslovak Mathematical Journal

For a graph G = ( V , E ) , a double Roman dominating function is a function f : V { 0 , 1 , 2 , 3 } having the property that if f ( v ) = 0 , then the vertex v must have at least two neighbors assigned 2 under f or one neighbor with f ( w ) = 3 , and if f ( v ) = 1 , then the vertex v must have at least one neighbor with f ( w ) 2 . The weight of a double Roman dominating function f is the sum f ( V ) = v V f ( v ) . The minimum weight of a double Roman dominating function on G is called the double Roman domination number of G and is denoted by γ dR ( G ) . In this paper, we establish a new upper bound...

A note on the independent domination number of subset graph

Xue-Gang Chen, De-xiang Ma, Hua Ming Xing, Liang Sun (2005)

Czechoslovak Mathematical Journal

The independent domination number i ( G ) (independent number β ( G ) ) is the minimum (maximum) cardinality among all maximal independent sets of G . Haviland (1995) conjectured that any connected regular graph G of order n and degree δ 1 2 n satisfies i ( G ) 2 n 3 δ 1 2 δ . For 1 k l m , the subset graph S m ( k , l ) is the bipartite graph whose vertices are the k - and l -subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i ( S m ( k , l ) ) and prove that...

A note on the independent domination number versus the domination number in bipartite graphs

Shaohui Wang, Bing Wei (2017)

Czechoslovak Mathematical Journal

Let γ ( G ) and i ( G ) be the domination number and the independent domination number of G , respectively. Rad and Volkmann posted a conjecture that i ( G ) / γ ( G ) Δ ( G ) / 2 for any graph G , where Δ ( G ) is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than Δ ( G ) / 2 are provided as well.

A note on the open packing number in graphs

Mehdi Mohammadi, Mohammad Maghasedi (2019)

Mathematica Bohemica

A subset S of vertices in a graph G is an open packing set if no pair of vertices of S has a common neighbor in G . An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by ρ o ( G ) . A subset S in a graph G with no isolated vertex is called a total dominating set if any vertex of G is adjacent to some vertex of S . The total domination number of G , denoted...

A Note on Total Graphs

S.F. Forouhandeh, N. Jafari Rad, B.H. Vaqari Motlagh, H.P. Patil, R. Pandiya Raj (2015)

Discussiones Mathematicae Graph Theory

Erratum Identification and corrections of the existing mistakes in the paper On the total graph of Mycielski graphs, central graphs and their covering numbers, Discuss. Math. Graph Theory 33 (2013) 361-371.

A remark on the (2,2)-domination number

Torsten Korneffel, Dirk Meierling, Lutz Volkmann (2008)

Discussiones Mathematicae Graph Theory

A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter γ k , p ( G ) denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γ k , p ( G ) ( p / ( p + k ) ) n ( G ) for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers...

A simple linear algorithm for the connected domination problem in circular-arc graphs

Ruo-Wei Hung, Maw-Shang Chang (2004)

Discussiones Mathematicae Graph Theory

A connected dominating set of a graph G = (V,E) is a subset of vertices CD ⊆ V such that every vertex not in CD is adjacent to at least one vertex in CD, and the subgraph induced by CD is connected. We show that, given an arc family F with endpoints sorted, a minimum-cardinality connected dominating set of the circular-arc graph constructed from F can be computed in O(|F|) time.

A survey on combinatorial optimization in dynamic environments∗

Nicolas Boria, Vangelis T. Paschos (2011)

RAIRO - Operations Research

This survey presents major results and issues related to the study of NPO problems in dynamic environments, that is, in settings where instances are allowed to undergo some modifications over time. In particular, the survey focuses on two complementary frameworks. The first one is the reoptimization framework, where an instance I that is already solved undergoes some local perturbation. The goal is then to make use of the information provided by the initial solution to compute a new solution. The...

A survey on combinatorial optimization in dynamic environments∗

Nicolas Boria, Vangelis T. Paschos (2011)

RAIRO - Operations Research

This survey presents major results and issues related to the study of NPO problems in dynamic environments, that is, in settings where instances are allowed to undergo some modifications over time. In particular, the survey focuses on two complementary frameworks. The first one is the reoptimization framework, where an instance I that is already solved undergoes some local perturbation. The goal is then to make use of the information provided by the initial solution to compute a new solution. The...

Currently displaying 21 – 40 of 51