Let G be a connected graph of order n, and let NC2(G) denote min{[N(u)UN(v)[: dist(u, v )= 2}, where dist(u, v) is the distance between u and v in G. A cycle C in G is called a dominating cycle, if V(G)\V(C) is an independent set in G. In this paper, we prove that if G contains a dominating cycle and ~ ~> 2, then G contains a dominating cycle of length at least min{n,2NC2(G)3}. ~ 1997 Elsevier ...

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...

The location of resources in a network satisfying some optimization property is a classical combinatorial problem that can be modeled and solved by using graphs. Key tools in this problem are the domination-type properties, which have been defined and widely studied in different types of graph models, such as undirected and directed graphs, finite and infinite graphs, simple graphs and hypergra...

A subset D of ( ) V G is called an equitable dominating set if for every ( ) v V G D   there exists a vertex u D  such that ( ) uv E G  and | ( ) ( ) | 1 deg u deg v   . A subset D of ( ) V G is called an equitable independent set if for any , u D v   ( ) e N u for all { } v D u   . The concept of equi independent equitable domination is a combination of these two important concepts. ...

It is shown that in star-free graphs the maximum independent set problem, the minimum dominating set problem and the minimum independent dominating set problem are approximable up to constant factor by any maximal independent set.

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.

A set S of vertices in a graph G is a 2-dominating set if every vertex of G not in S is adjacent to at least two vertices in S, and S is a 2-independent set if every vertex in S is adjacent to at most one vertex of S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set in G, and the 2-independence number α2(G) is the maximum cardinality of a 2-independent set in G. Ch...

A Roman dominating function of a graph G = (V,E) is a function f : V → {0, 1, 2} such that every vertex x with f(x) = 0 is adjacent to at least one vertex y with f(y) = 2. The weight of a Roman dominating function is defined to be f(V ) = P x∈V f(x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we answer an open pro...

We first devise a branching algorithm that computes a minimum independent dominating set with running time O∗(1.3351n) = O∗(20.417n) and polynomial space. This improves upon the best state of the art algorithms for this problem. We then study approximation of the problem by moderately exponential time algorithms and show that it can be approximated within ratio 1 + ε, for any ε > 0, in a time s...

Rainbows are a natural cue for calibrating outdoor imagery. While ephemeral, they provide unique calibration cues because they are centered exactly opposite the sun and have an outer radius of 42 degrees. In this work, we define the geometry of a rainbow and describe minimal sets of constraints that are sufficient for estimating camera calibration. We present both semi-automatic and fully autom...