In this paper, we consider theMinimum Independent Dominating Set problem and develop exact exponential algorithms that break the trivial O(2|V |) bound. A simple O∗( √ 3 |V | ) time algorithm is developed to solve this problem on general graphs. For sparse graphs, e.g. graphs with degree bounded by 3 and 4, we show that a few new branching techniques can be applied to these graphs and the resul...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

The construction of a virtual backbone for ad hoc networks is modelled by connected dominating set (CDS) in unit-disk graphs. This paper introduces a novel idea to compute CDS effectively our algorithm does not include the members of an existing maximal independent set (MIS) even though it does connect the MIS. Popular approaches in literature count the MIS in the resultant CDS. Therefore it is...

We provide self-stabilizing algorithms to obtain and maintain a maximal matching, maximal independent set or minimal dominating set in a given system graph. They converge in linear rounds under a distributed or synchronous daemon. They can be implemented in an ad hoc network by piggy-backing on the beacon messages that nodes already use.

The domination number γ(G) of the fuzzy graph G is the minimum cardinality taken over all minimal dominating sets of G. The independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of G. The irredundant number ir(G) is the minimum cardinality taken over all maximal irredundant sets of G. In this paper we prove the result that relate the parameters ir...

This paper contains a number of estimations of the split domination number and the maximal domination number of a graph with a deleted subset of edges which induces a complete subgraph Kp. We discuss noncomplete graphs having or not having hanging vertices. In particular, for p = 2 the edge deleted graphs are considered. The motivation of these problems comes from [2] and [6], where the authors...