Induced Matchings in Subcubic Planar Graphs

Abstract

Abstract. We present a linear-time algorithm that, given a planar graph with m edges and maximum degree 3, finds an induced matching of size at least m/9. This is best possible.

Strong chromatic index of subcubic planar multigraphs

“…Theorem 1 implies the following corollary. This corollary extends a result of Kang, Mnich and Müller [16] to loopless multigraphs. Joos, Rautenbach and Sasse [15] later showed that the above lower bound holds for all subcubic graphs, thus proving a conjecture of Henning and Rautenbach [10].…”

Induced Matchings in Subcubic Graphs

“…Theorem 2 actually implies the main result of Kang, Mnich, and Müller, Theorem 1 in [10], stating that every subcubic planar graph G of size m(G) has an induced matching of size at least m(G)/9, which can be determined in linear time. In fact, since K + 3,3 is not planar, every subcubic planar graph G satisfies n + 3,3 (G) = 0.…”

“…Our proof of Theorem 2 is constructive and leads to a linear time algorithm to determine an induced matching of the guaranteed size as we will explain below. While the proof in [10] is extremely involved, our proof of Theorem 2 is quite simple. One reason for this simplicity is probably the fact that we express the lower bound on the strong matching number in terms of the order.…”

“…One reason for this simplicity is probably the fact that we express the lower bound on the strong matching number in terms of the order. Since the strong matching number is the cardinality of an edge set, it might seem more natural to express this lower bound in terms of the size as in [10]. Nevertheless, edges that are incident with low degree vertices should contribute more to the strong matching number and our lower bound that is essentially linear in the order implicitly captures this intuitive idea.…”

Induced Matchings in Graphs of Bounded Maximum Degree

“…For subcubic planar graphs, Kang, Mnich and Müller [9] showed that ν s (G) ≥ m(G) 9 . This was improved by Rautenbach et al [7]…”