A sequence of vertices in a graph G is called a legal dominating sequence if every vertex in the sequence dominates at least one vertex not dominated by those vertices that precede it, and at the end all vertices of G are dominated. While the length of a shortest such sequence is the domination number of G, in this paper we investigate legal dominating sequences of maximum length, which we call...

Let γ ′ s (G) be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n ≥ 2), γ ′ s (G) ≥ 1. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that γ ′ s (G) ≤ − m 6 |V (G)|. Also for every two natural numbers m and n, we determine γ ′...

Abstract In this work, we study the following problem: given a connected graph G , can reduce domination number of by at least one using k edge contractions, for some fixed integer > 0 ? We show that = 1 (resp.  2 ), problem is NP -hard coNP -hard). further prove W [1]-hard parameterized plus mim-width input graph, and it remains when restricted to chordal { P 6 4 + } -free graphs, bipartite gr...

For a graphical property P and a graph G, a subset S of vertices of G is a P-set if the subgraph induced by S has the property P . The domination number with respect to the property P , is the minimum cardinality of a dominating P-set. In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate and hereditary properties when a graph is ...

For an integer n ≥ 2, let I ⊂ {0, 1, 2, · · · , n}. A Smarandachely Roman sdominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a function f : V → {0, 1, 2, · · · , n} satisfying the condition that |f(u)− f(v)| ≥ s for each edge uv ∈ E with f(u) or f(v) ∈ I . Similarly, a Smarandachely Roman edge s-dominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a func...

for a graph $g$ let $gamma (g)$ be its domination number. we define a graph g to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ed}$ graph) if $g$ has no efficient dominating set (eds) but every graph formed by removing a single vertex from $g$ has at least one eds, and (ii) a hypo-unique domination graph (a hypo-$mathcal{ud}$ graph) if $g$ has at least two minimum dominating sets...

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

The inflated graph GI of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj , and two different edges of G are replaced by non-adjacent edges of GI . For integer k ≥ 1, the k-tuple total domination number γ ×...