Finding Large Matchings in 1-Planar Graphs of Minimum Degree 3

Abstract

In this paper, we study the maximum matching problem in RDV graphs, i.e., graphs that are vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a vertical segment intersects one of a dynamically changing set of horizontal segments, which in turn reduces to an orthogonal ray shooting query. Using a suitable data structure, we can therefore find a maximum matching in O(n log n) time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.

Matchings in 1‐planar graphs with large minimum degree

“…≥ could be shown with much the same proof. We will at the same time prove another result that is related (but neither result implies the other); this is needed in another paper [4]. Define the crossing-weighted degree of a vertex v V ∈ to be the degree of v plus the number of incident uncrossed edges at v. Thus, uncrossed edges count doubly.…”

On the size of matchings in 1-planar graph with high minimum degree

“…Each good 5-vertex-component F of G\S exactly contributes 5 vertices to T and at least 12 edges to G * , because |E G (V (F ), S)| ≥ 12 by the definition of good 5-vertex-components of G\S.Fact (4). Each bad 5-vertex-component F of G\S with G (v i , S) ≥ 4 exactly contributes 2 vertices to T and at least 6 edges to G * by(9).Fact(5).Each bad 5-vertex-component F of G\S with G (v i , S) = 3 and N G (v 0 , S) ⊆ G (v i , S) exactly contributes one vertex to T and at least 4 edges to G * by(10).Fact(6). Each bad 5-vertex-component F of G\S with4 i=1 N G (v i , S) = 3 and N G (v 0 , S) ⊆ G (v i , S) exactly contributes 3 vertices to T and at least 8 edges to G * by (11).…”