A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

‎Let $G=(V(G),E(G))$ be a graph‎, ‎$gamma_t(G)$. Let $ooir(G)$ be the total domination and OO-irredundance number of $G$‎, ‎respectively‎. ‎A total dominating set $S$ of $G$ is called a $textit{total perfect code}$ if every vertex in $V(G)$ is adjacent to exactly one vertex of $S$‎. ‎In this paper‎, ‎we show that if $G$ has a total perfect code‎, ‎then $gamma_t(G)=ooir(G)$‎. ‎As a consequence, ...

a subset $s$ of vertices in a graph $g$ is called a geodetic set if every vertex not in $s$ lies on a shortest path between two vertices from $s$‎. ‎a subset $d$ of vertices in $g$ is called dominating set if every vertex not in $d$ has at least one neighbor in $d$‎. ‎a geodetic dominating set $s$ is both a geodetic and a dominating set‎. ‎the geodetic (domination‎, ‎geodetic domination) number...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...

Let G = (V,E) be a graph. A subset D ⊆ V is a dominating set if every vertex not in D is adjacent to a vertex in D. A dominating set D is called a total dominating set if every vertex in D is adjacent to a vertex in D. The domination (resp. total domination) number of G is the smallest cardinality of a dominating (resp. total dominating) set of G. The bondage (resp. total bondage) number of a n...

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are ob...

Abstract Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to double total dominating if $$|N(v)\cap D|\ge 2$$ | N ...

Consider a simple connected fuzzy graph (FG) G and consider an ordered subset H = {(u1, σ(u1)), (u2, σ(u2)), …(uk, σ(uk))}, |H| ≥ 2 of graph; then, the representation σ − is k-tuple with regard to G. If any two elements do not have distinct H, then this called resolving set (FRS) smallest cardinality known as number (FRN) it denoted by Fr(G). Similarly, S such that for u∈S, ∃v∈V S, dominating o...