let $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

We investigate the minimum independent dominating set in perturbed graphs g(G, p) of input graph G = (V,E), obtained by negating the existence of edges independently with a probability p > 0. The minimum independent dominating set (MIDS) problem does not admit a polynomial running time approximation algorithm with worst-case performance ratio of n1− for any > 0. We prove that the size of the mi...

Connected dominating set is widely used in wireless ad-hoc and sensor networks as a routing and topology control backbone to improve the performance and increase the lifetime of the network. Most of the distributed algorithms for approximating connected dominating set are based on constructing maximal independent set. The performance of such algorithms highly depends on the relation between the...

A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) to {0, 1, 2}$satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least onevertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independentRoman dominating function (OIRDF) on $G$ if the set ${vin Vmid f(v)=0}$ is independent.The (outer-independent) Roman dom...

We introduce a surprisingly simple technique to design and analyze algorithms based on search trees, that significantly improves many existing results in the area of exact algorithms. The technique is based on measuring the progress of Branch & Bound algorithms by making use of a combinatorial relation between the average and maximum dual degrees of a graph. By dual degree of a vertex, we mean ...

An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called AT-free if it does not have an AT. We show that there is an O(n4) time algorithm to compute the maximum weight of an independent set for AT-free graphs. Furthermore, we obtain O(n4) time algorithms to solve the independent domin...

In a graph G, an efficient dominating set is a subset D of vertices such that D is an independent set and each vertex outside D has exactly one neighbor in D. The Efficient Dominating Set problem (EDS) asks for the existence of an efficient dominating set in a given graph G. The EDS is known to be NP -complete for P7-free graphs, and is known to be polynomial time solvable for P5-free graphs. H...

Let S be a bipartite semigraph with |NXa(y)| ≥ 1 for every y ∈ Y . A vertex x ∈ X hyper dominates y ∈ Y if y ∈ Na(x) or y ∈ Na(NY a(x)). A subsetD ⊆ X is a hyper dominating set of S if every y ∈ Y is hyper dominated by a vertex ofD. A subsetD ⊆ X is called a minimal hyper dominating set of S if no proper subset of D is a hyper dominating set of S. The minimum cardinality of a minimal hyper domi...

In this paper, we introduce a new scheme of parameters which generalises many wellknown parameters, including the domination number, the total domination number, the independence number, and the vertex covering number.

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we offer a survey of selected recent results on independent domination in graphs.