Let G = (V, E) be a graph. A set M of edges is called a matching in G if each vertex in G belongs to at most one edge from M , and M is a maximal matching if any edgeset M ′, such that M ⊂ M ′, is not a matching in G. If all maximal matchings in G have the same cardinality then G is an equimatchable graph. In this paper we characterize the equimatchable graphs of girth at least five. As a conse...

a {em roman dominating function} on a graph $g$ is a function$f:v(g)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}a {em restrained roman dominating}function} $f$ is a {color{blue} roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} the wei...

A paired-dominating set of a graph is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph. Recently, Chen, Sun and Xing [Acta Mathematica Scientia Series A Chinese Edition 27(1) (2007), 166–170] proved that a cubic graph has paired-domination number at most three-fifths ...

This paper studies Upper Domination, i.e., the problem of computing the maximum cardinality of a minimal dominating set in a graph, with a focus on parameterised complexity. Our main results include W[1]-hardness for Upper Domination, contrasting FPT membership for the parameterised dual Co-Upper Domination. The study of structural properties also yields some insight into Upper Total Domination...

The nonclassical mixed domination Ramsey number v(m,G) is the smallest integer p such that in every 2-coloring of the edges of Kp with color red and blue, either Γ(B) ≥ m or there exists a blue copy of graph G, where B is the subgraph of Kp induced by blue edges. Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. We give exact values for numbers v(m,K3 − e), v(3, Pm), v(3...

In a graph or network G = (V;E), a set S V is k-dependent if no node in S has more than k neighbors in S. We show that for each k 0 there is a self-stabilizing algorithm that identi es a maximal k-dependent set, and stabilizes in O(m + n) time. An interesting by-product of our paper is the new result that in any graph there exists a set that is both maximal k-dependent and minimal (k+1)-dominat...

Let G = (V,E) be a graph and let v ∈ V. Let γ(v,G) denote the minimum cardinality of a minimal dominating set of G containing v. Then γ(G) = max{γ(v,G) : v ∈ V (G)} is called the min-max dom-saturation number of G. In this paper we present a dynamic programming algorithm for determining the min-max domsaturation number of a tree.

The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kp with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kp induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4, 4) ≤ 15.

The domination number γ(G) of the fuzzy graph G is the minimum cardinality taken over all minimal dominating sets of G. The independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of G. The irredundant number ir(G) is the minimum cardinality taken over all maximal irredundant sets of G. In this paper we prove the result that relate the parameters ir...

Let m and n be positive integers. The algorithmic search for perfect dominating sets of the rectangular grid Gm;n satisfying an initial condition S 0 de ned as an admissible subset of a side Gm;1 of Gm;n such that S \ Gm;1 = S 0 is considered. A binary decision algorithm that generates all such perfect dominating sets is presented, and some related questions and conjectures are posed, leading t...