1 Copyright © 2016, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. Chapter 1 DOI: 10.4018/978-1-4666-9964-9.ch001 ABSTRACT An undirected graph can be represented by G(V,E) where V is the set of vertices and E is the set of edges connecting vertices. The problem of fnding a vertex cover (VC) is to identify a set of vertices VC such that at least one endpoint of every edge in E is incident to a vertex V in VC. Vertex cover is a very important graph theoretical structure for various types of communication networks such as wireless sensor networks, since VC can be used for link monitoring, clustering, backbone formation and data aggregation management. In this chapter, we will defne vertex cover and related problems with their applications on communication networks and we will survey some important distributed algorithms on this research area. INTRODUCTION In this section the vertex cover problems and the contributions of this chapter will be introduced. Vertex Cover Problems Given a graph G(V, E) where V is the set of vertices and E is the set of edges between vertices, the problem to find a set of vertices VC ∈ V such that for any edge {u,v} ∈ E, at least one of u and v is in VC is called vertex cover problem. V itself is a vertex cover and it may have numerous subsets satisfying the vertex cover conditions. Among all possible vertex covers of a given graph, the one(s) that have the minimum cardinality are called the minimum vertex cover. Finding the minimum vertex cover on an undirected graph is an NP-Hard problem. A minimal vertex cover is a vertex cover VC whose cardinal- ity cannot be decreased, in other words, exclusion of any vertex from VC would break the vertex cover On Vertex Cover Problems in Distributed Systems Can Umut Ileri Ege University, Turkey Cemil Aybars Ural Ege University, Turkey Orhan Dagdeviren Ege University, Turkey Vedat Kavalci Izmir University, Turkey