Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V \ S is adjacent to a vertex in S as well as to another vertex in V \S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ r (G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper boun...

Constructing a connected dominating set as the virtual backbone plays an important role in wireless networks. In this paper, we propose two novel approximate algorithms for dominating set and connected dominating set in wireless networks, respectively. Both of the algorithms are based on edge dominating capability which is a novel notion proposed in this paper. Simulations show that each of pro...

A set of vertices is a dominating set in a graph if every vertex not in the dominating set is adjacent to one or more vertices in the dominating set. A dominating clique is a dominating set that induces a complete subgraph. Forbidden subgraph conditions sufficient to imply the existence of a dominating clique are given. For certain classes of graphs, a polynomial algorithm is given for finding ...

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. In terms of a chess board problem, let Xn be the graph for chess pieceX on the square of side n. Thus, (Xn) is the domination number for che...

∗The authors thank the financial support received from Grant UNAM-PAPIIT IN114415 and SEP-CONACyT. Also, the first author would like to thank the support of the Post-Doctoral Fellowships program of DGAPA-UNAM. †email: [email protected] (Corresponding Author) ‡[email protected] §[email protected] 1 ar X iv :1 70 5. 00 21 6v 1 [ m at h. C O ] 2 9 A pr 2 01 7 A vertex cover of a grap...

‎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, ...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. A set S ⊆ V is a restrained dominating set if every vertex in V − S is adjacent to a vertex in S and to a vertex in V − S. The total restrained domination number of G (restrained domination number of G, respectively),...

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...

For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V (D) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (D), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) consists of all vertices of D from which arcs go into v. A set {f1, f2, . . . , fd} of total {k}-dominating functions of D with the property that ∑ d i=1 fi(...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex in V is adjacent to a vertex in S and every vertex of V −S is adjacent to a vertex in V −S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a total restrained dominating set of G. A unicyclic graph is a connected graph that contains precisely one cycle. We sho...