Vertex Cover Gets Faster and Harder on Low Degree Graphs

TL;DR

This paper investigates the complexity of the vertex cover problem on low-degree graphs, especially those formed by the union of two Hamiltonian paths, and presents an algorithm with fixed-parameter tractability.

Contribution

It establishes NP-completeness for vertex cover on certain low-degree graphs and provides a fixed-parameter algorithm for graphs with maximum degree four.

Findings

Vertex cover on graphs formed by two Hamiltonian paths is NP-complete.

An algorithm with runtime 1.2637^k n^O(1) for vertex cover on degree-4 graphs.

Connection between hitting geometric objects and vertex cover on special graph classes.

Abstract

The problem of finding an optimal vertex cover in a graph is a classic NP-complete problem, and is a special case of the hitting set question. On the other hand, the hitting set problem, when asked in the context of induced geometric objects, often turns out to be exactly the vertex cover problem on restricted classes of graphs. In this work we explore a particular instance of such a phenomenon. We consider the problem of hitting all axis-parallel slabs induced by a point set P, and show that it is equivalent to the problem of finding a vertex cover on a graph whose edge set is the union of two Hamiltonian Paths. We show the latter problem to be NP-complete, and we also give an algorithm to find a vertex cover of size at most k, on graphs of maximum degree four, whose running time is 1.2637^k n^O(1).