Abstract In this paper, we study the problem of deciding whether total domination number a given graph G can be reduced using exactly one edge contraction (called 1 -Edge Contraction( ? t ) ). We focus on several classes and determine computational complexity problem. By putting together these results, manage to obtain complete dichotomy for H-free graphs.

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequen...

An {em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function$fcolon V(D)to {0, 1, 2}$ such that every vertex $vin V(D)$ with $f(v)=0$ has at least two in-neighborsassigned 1 under $f$ or one in-neighbor $w$ with $f(w)=2$. A set ${f_1,f_2,ldots,f_d}$ of distinctItalian dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vi...

In this paper, we define a new domination-like invariant of graphs. Let $${\mathbb {R}}^{+}$$ be the set non-negative numbers. $$c\in {\mathbb {R}}^{+}-\{0\}$$ number, and let G graph. A function $$f:V(G)\rightarrow is c-self-dominating if for every $$u\in V(G)$$ , $$f(u)\ge c$$ or $$\max \{f(v):v\in N_{G}(u)\}\ge 1$$ . The c-self-domination number $$\gamma ^{c}(G)$$ defined as ^{c}(G):=\min \{...

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any in...

For a graph [Formula: see text], double Roman dominating function (DRDF) is text] having the property that if for some vertex then has at least two neighbors assigned under or one neighbor with and text]. The weight of DRDF sum minimum on domination number denoted by bondage cardinality among all edge subsets such In this paper, we study in graphs. We determine several families graphs, present ...