Wedesign fast exponential time algorithms for some intractable graph-theoretic problems. Ourmain result states that aminimum optional dominating set in a graph of order n can be found in time O∗(1.8899n). Ourmethods to obtain this result involvematching techniques. The list of the considered problems includes Minimum Maximal Matching, 3Colourability, Minimum Dominating Edge Set, Minimum Connect...

Let G be a connected graph and S ⊆ V (G). Then the Steiner distance of S, denoted by dG(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I (S) is the union of all vertices that belong to some Steiner tree for S. If S = {u, v}, then ...

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset E of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E. We say that E is a perfect edge dominating set of G, if every edge not in E is dominated by exactly one edge of E. The perfect edge dominating problem is to determine a least cardinality perf...

This contribution reviews a few basic concepts of optimization and design of a geodetic network. Proper assessment and analysis of networks is an important task in many geodetic-surveying projects. Appropriate quality-control measures should be defined, and an optimal design should be sought. The quality of a geodetic network is characterized by precision, reliability, and cost. The aim is to p...

Let G = (V, E) be a graph without isolated vertices. A secure edge dominating set of G is an edge dominating set F⊆E with the property that for each e ∈ E – F, there exists f∈F adjacent to e such that (F – {f}) ∪ {e} is an edge dominating set. The secure edge domination number γ's(G) of G is the minimum cardinality of a secure edge dominating set of G. In this paper, we initiate a study of the ...

Given a connected graph and a vertex x ∈ V (G), the geodesic graph Px(G) has the same vertex set as G with edges uv iff either v is on an x − u geodesic path or u is on an x − v geodesic path. A characterization is given of those graphs all of whose geodesic graphs are complete bipartite. It is also shown that the geodetic number of the Cartesian product of Km,n with itself, where m,n ≥ 4, is e...

let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

Let G = (V; E) be a /nite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E \ D is dominated by exactly one edge in D and an e cient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (e cient) edge domination prob...

A set \(S\) of vertices in a connected graph \(G=(V,E)\) is called signal if every vertex not lies on path between two from \(S\). double \(G\) for each pair \(x,y \in G\) there exist \(u,v S\) such that L[u,v]\). The number \(\mathrm{dsn}\,(G)\) the minimum cardinality set. Any \(\mathrm{dsn}\)-set \(G\). In this paper we introduce and initiate some properties graph. We have also given relatio...