a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

Let G be an undirected graph with vertex and edge sets V (G) E(G), respectively. A subset S of vertices is a geodetic hop dominating set if it both set. The domination number G, γhg(G), the minimum cardinality among all in G. Geodetic resulting from some binary operations have been characterized. These characterizations used to determine tight bounds for each graphs considered.

Given a Graph G = ((V(G),E(G)), and a subset ) (G V S  , S with a given property(covering set, Dominating set, Neighbourhood set), we define a matrix taking a row for each of the minimal set corresponding to the given property and a column for each of the vertex of G. The elements of the matrix are 1 or 0 respectively as the vertex is contained in minimal set or otherwise. That is matrix (mij)...

A function f :V (G) → {−1, 0, 1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V (G) → {−1, 0, 1}, f = g, for which g(v) f (v) for every v ∈ V (G). The weight of an MTDF is the sum of its function values over all vertices. The mi...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

A dominating set in a graph G = ( V, E ) is S such that every vertex of either or adjacent to . While the minimum cardinality called domination number denoted by Γ( ), maximum minimal upper ). We call difference between these two parameters gap and denote it µ d − with 0 said be well-dominated graph, we 1 an almost graph. In this work, first establish bound for bipartite graphs k , where ≥ 1, d...

A weakly-connected dominating set, W, of a graph, G, is a dominating set such that the subgraph consisting of V (G) and all edges incident with vertices in W is connected. Finding a small weaklyconnected dominating set of a graph has important applications in clustering mobile ad-hoc networks. In this paper we introduce several new randomised greedy algorithms for finding small weakly-connected...