On the complexity of finding a spanning even tree in a graph

TL;DR

This paper investigates the computational complexity of finding spanning even trees in graphs, revealing NP-completeness in general but providing polynomial algorithms for specific graph classes.

Contribution

It proves NP-completeness for planar graphs and offers polynomial-time solutions for several restricted graph classes.

Findings

NP-complete for planar graphs

Polynomial algorithms for split, cograph, cobipartite, unit interval, and block graphs

Extends understanding of the problem's complexity landscape

Abstract

A tree is said to be even if for every pair of distinct leaves, the length of the unique path between them is even. In this paper we discuss the problem of determining whether an input graph has a spanning even tree. Hofmann and Walsh [Australas. J Comb. 35, 2006] proved that this problem can be solved in polynomial time on bipartite graphs. In contrast to this, we show that this problem is NP-complete even on planar graphs. We also give polynomial-time algorithms for several restricted classes of graphs, such as split graphs, cographs, cobipartite graphs, unit interval graphs, and block graphs.