Connected and Total Vertex covering in Graphs

A Subset S of vertices of a Graph G is called a vertex cover if S includes at least one end point of every edge of the Graph. A Vertex cover S of G is a connected vertex cover if the induced subgraph of S is connected. The minimum cardinality of such a set is called the connected vertex covering number and it is denoted by . A Vertex cover S of G is a total vertex cover if the induced subgraph of S has no isolates. The minimum cardinality of such a set is called the total vertex covering number and it is denoted by .In this paper a few properties of connected vertex cover and total vertex covers are studied and specific values of and of some well-known graphs are evaluated.

An Efficient Heuristic Algorithm for Solving Connected Vertex Cover Problem

The connected vertex cover (CVC) problem, which has many important applications, is a variant of the vertex cover problem, such as wireless network design, routing, and wavelength assignment problem. A good algorithm for the problem can help us improve engineering efficiency, cost savings, and resources consumption in industrial applications. In this work, we present an efficient algorithm GRASP-CVC (Greedy Randomized Adaptive Search Procedure for Connected Vertex Cover) for CVC in general graphs. The algorithm has two main phases, i.e., construction phase and local search phase. In the construction phase, to construct a high quality feasible initial solution, we design a greedy function and a restricted candidate list. In the local search phase, the configuration checking strategy is adopted to decrease the cycling problem. The experimental results demonstrate that GRASP-CVC is better than other comparison algorithms in terms of effectivity and efficiency.

Layered Graphs: A Class that Admits Polynomial Time Solutions for Some Hard Problems

The independent set, IS, on a graph G = ( V , E ) is V * ⊆ V such that no two vertices in V * have an edge between them. The MIS problem on G seeks to identify an IS with maximum cardinality, i.e. MIS. V * ⊆ V is a vertex cover, i.e. VC of G = ( V , E ) if every e ∈ E is incident upon at least one vertex in V * . V * ⊆ V is dominating set, DS, of G = ( V , E ) if ∀ v ∈ V either v ∈ V * or ∃ u ∈ V * and ( u , v ) ∈ E . The MVC problem on G seeks to identify a vertex cover with minimum cardinality, i.e. MVC. Likewise, MCV seeks a connected vertex cover, i.e. VC which forms one component in G, with minimum cardinality, i.e. MCV. A connected DS, CDS, is a DS that forms a connected component in G. The problems MDS and MCD seek to identify a DS and a connected DS i.e. CDS respectively with minimum cardinalities. MIS, MVC, MDS, MCV and MCD on a general graph are known to be NP-complete. Polynomial time algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs for some of these problems. We introduce a novel graph class, layered graph, where each layer refers to a subgraph containing at most some k vertices. Inter layer edges are restricted to the vertices in adjacent layers. We show that if k = Θ ( log ∣ V ∣ ) then MIS, MVC and MDS can be computed in polynomial time and if k = O ( ( log ∣ V ∣ ) α ) , where α < 1 , then MCV and MCD can be computed in polynomial time. If k = Θ ( ( log ∣ V ∣ ) 1 + ϵ ) , for ϵ > 0 , then MIS, MVC and MDS require quasi-polynomial time. If k = Θ ( log ∣ V ∣ ) then MCV, MCD require quasi-polynomial time. Layered graphs do have constraints such as bipartiteness, planarity and acyclicity.