Vector connectivity in graphs1. Issue 4 (10th February 2014)

• Record Type: Journal Article
• Authors: Boros, Endre; Heggernes, Pinar; van 't Hof, Pim; Milanič, Martin
• Is Part Of: Networks. Volume 63:Issue 4(2014:Jul.)
• Page Start: 277
• Page End: 285
• Publication Date: 2014-02-10
• Subjects: vector connectivity -- approximation algorithm -- polynomial time algorithm -- split graph -- cograph -- tree; Network analysis (Planning) -- Periodicals; 658.4032
• DOI: 10.1002/net.21545
• Languages: English
• ISSNs: 0028-3045

Abstract

Abstract: Motivated by challenges related to domination, connectivity, and information propagation in social and other networks, we initiate the study of theVECTOR CONNECTIVITY problem. This problem takes as input a graph G and an integer k v for every vertex v of G, and the objective is to find a vertex subset S of minimum cardinality such that every vertex v either belongs to S, or is connected to at least k v vertices of S by disjoint paths. If we require each path to be of length exactly 1, we get the well‐knownVECTOR DOMINATION problem, which is a generalization of the famousDOMINATING SET problem and several of its variants. Consequently, our problem becomes NP‐hard if an upper bound on the length of the disjoint paths is also supplied as input. Due to the hardness of these domination variants even on restricted graph classes, like split graphs, VECTOR CONNECTIVITY seems to be a natural problem to study for drawing the boundaries of tractability for this type of problems. We show thatVECTOR CONNECTIVITY can actually be solved in polynomial time on split graphs, in addition to cographs and trees. We also show that the problem can be approximated in polynomial time within a factor of ln n + 2 on all n ‐vertex graphs.Copyright © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 63(4), 277–285 2014