Abstract
Graph Theory The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the \em Price of Connectivity, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number t. We obtain forbidden induced subgraph characterizations for every real value t ≤q 3/2. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most t. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class Θ₂^P = P^NP[\log], the class of decision problems that can be solved in polynomial time, provided we can make O(\log n) queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.
Highlights
A vertex cover of a graph G is a vertex subset C such that every edge of G has at least one endpoint in C
A well-known variant of the notion of vertex cover is that of connected vertex cover, defined as a vertex cover Cc such that the induced subgraph G[Cc] is connected. (If G is not connected we ask that G[Cc] has the same number of component as G.) The minimum size of such a set, denoted by τc(G), is the connected vertex cover number of G
Let us first note that every vertex cover C of a connected graph G such that G[C] has c connected components can be turned into a connected vertex cover of G by adding at most c − 1 vertices
Summary
A vertex cover of a graph G is a vertex subset C such that every edge of G has at least one endpoint in C. Fernau and Manlove [5] showed th√at the connected vertex cover problem is not approximable within an asymptotic performance ratio of 10 5 − 21 − δ, for any δ > 0, unless P = N P. Let us first note that every vertex cover C of a connected graph G such that G[C] has c connected components can be turned into a connected vertex cover of G by adding at most c − 1 vertices The Price of Connectivity (as defined here) has been introduced by Cardinal and Levy [3, 8], who showed that it was bounded by 2/(1 + ε) in graphs with average degree εn, where n denotes the number of vertices. Fulman [7] and Zverovich [15] investigated the ratio between the independence number and the upper domination number
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
