The Generalized Minimum Edge Biconnected Network Problem (GMEBCNP) is defined as follows. We consider an undirected weighted graph G = 〈V, E, c〉 with node set V , edge set E, and edge cost function c : E → R. Node set V is partitioned into r pairwise disjoint clusters V1, V2, . . . , Vr, ⋃r i=1 Vi = V, Vi ∩ Vj = ∅ ∀i, j = 1, . . . , r, i 6= j. A solution to the GMEBCNP defined on G is a subgrap...

A subset X of edges of a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X . The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} such that every edge e with f(e) = 0 is adjacent to some edge e′ with f(e′) = 2. T...

Given a set P of n points in the plane, we solve the problems of constructing a geometric planar graph spanning P 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set P, respectively. First, we construct in O(...

An edge dominating set of a graph G = (V,E) is a subset M ⊆ E of edges such that each edge in E \ M is incident to at least one edge in M . In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O(2.2351)-time and polynomial-space algorit...

The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...

Let G = (V,E) be a weighted undirected graph with vertex set V , edge set E and a weight function d. Thus, d(u, v) denotes the weight of any edge (u, v) ∈ E. A matching M ⊆ E is a collection of edges such that every node in V is incident to at most one edge in M . The matching is perfect if every node in V is incident to exactly one edge in M . The cost of the matching is given by ∑ (u,v)∈M d(u...

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio...