Lower Bounds for Domination and Total Domination Number of Direct Products Graphs
نویسنده
چکیده
An exact lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: (×i=1Kni) ≥ t + 1, t ≥ 3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: (G×H) ≥ (G)+ (H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds t(G×H) = (G) + (H)− 1. Some additional parallels with the total domination number are made.
منابع مشابه
More results on total domination in direct products of graphs
Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number and the total 2-tuple domination number of the factors. Using these relationships some exact total domination numbers are obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direc...
متن کاملSome results on total domination in direct products of graphs
Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The dom...
متن کاملLower bounds on the signed (total) $k$-domination number
Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...
متن کاملTotal Roman domination subdivision number in graphs
A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...
متن کاملPacking and Domination Invariants on Cartesian Products and Direct Products
The dual notions of domination and packing in finite simple graphs were first extensively explored by Meir and Moon in [15]. Most of the lower bounds for the domination number of a nontrivial Cartesian product involve the 2-packing, or closed neighborhood packing, number of the factors. In addition, the domination number of any graph is at least as large as its 2-packing number, and the invaria...
متن کامل