Maximum k-regular induced subgraphs

Abstract

Abstract. Independent sets, induced matchings and cliques are examples of regular induced subgraphs in a graph. In this paper, we prove that finding a maximum cardinality k-regular induced subgraph is an NP-hard problem for any value of k. We propose a convex quadratic upper bound on the size of a k-regular induced subgraph and characterize those graphs for which this bound is attained. Finally, we extend the Hoffman bound on the size of a maximum 0-regular subgraph (the independence number) from k = 0 to larger values of k.

Induced Cycles in Graphs

“…We note that the vertex v ∈ V (G) has degree 4 in H , the vertex v has degree 3 in H , while each vertex of P v has degree 2 in H . Let C v be the induced cycle containing P v in H , and so C v is given by vv 1 …”

“…Cardoso et al [1] showed that it is NP-hard to find a maximum induced r -regular subgraph of a given graph. Lozin et al [7] established efficient algorithms for special graph classes including 2P 3 -free graphs, while Moser and Thilikos [8] studied FPT-algorithms for finding regular induced subgraphs.…”

“…Let P be the colored path in H that corresponds to the colored edge e i . (a) If P contains at least two yellow vertices, then remove all vertices of P from I1 and I 2 and we decide thate i / ∈ M i . This implies that |C ∩ h(v)| ≥ 2 3 n(h(v)) for every v ∈ V (C), and therefore that v∈V (C) a(v) ≥ 0.…”

Inapproximability of Maximum <i>r</i>-Regular Induced Connected Subgraph Problems

“…The parameterized complexity and exact exponential algorithms of r-MaxRIS are studied in [9] and [5], respectively. Very recently, in [2] Cardoso, Kamińsi, and Lozin prove that rMaxRIS is NP-hard for any value of r ≥ 3. Motivated by this result, we investigate the complexity of the connected version problem r-MaxRICS for r ≥ 3 in this paper.…”

Spectral Characterization of Families of Split Graphs

“…A convex quadratic upper bound on the order of k-regular induced subgraphs was obtained in [1] (see also [2]). In Section 3, for some connected graphs, a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set is introduced.…”