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

TL;DR

The paper analyzes the problem of deciding whether a graph has a spanning even tree, establishing NP-hardness on planar graphs while achieving polynomial-time solutions for several structured classes. It leverages SAT-based reductions and Planar 3SAT to prove hardness, and develops class-specific algorithms that exploit structural decompositions and colorings to construct spanning even trees when possible. The key contributions are a clear boundary between intractable and tractable cases, plus linear-time constructions for cographs, cobipartite graphs, unit interval graphs, split graphs, and block graphs. Together, these results guide algorithm design for parity-constrained spanning trees in diverse graph families and stimulate further study of related graph classes and degree-restricted cases.

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.

Figures (7)

• Figure 1: The left figure depicts the variable gadget $G_i$. There are two ways to take a spanning even tree in $G_i$.
• Figure 2: The figures illustrate the (partial) graph for $\varphi = (\neg x_1 \lor \neg x_2 \lor x_4) \land (x_1 \lor \neg x_2 \lor x_4) \land (x_2 \lor x_3 \lor \neg x_4)$ and the subgraph constructed from the assignment $(x_1, x_2, x_3, x_4) = (\texttt{false}, \texttt{false}, \texttt{true}, \texttt{true})$.
• Figure 3: A garbage collector gadget and a forest that has no white leaves.
• Figure 4: The left figure depicts the connector gadget $H_i$.
• Figure 5: The figures illustrate the entire graph $G_\varphi$ for $\varphi = (\neg x_1 \lor \neg x_2 \lor x_4) \land (\neg x_1 \lor \neg x_2 \lor x_4) \land (x_2 \lor x_3 \lor \neg x_4)$ and a spanning even tree obtained from the assignment $(x_1, x_2, x_3, x_4) = (\texttt{false}, \texttt{false}, \texttt{true}, \texttt{true})$.

Theorems & Definitions (20)

• Theorem 1
• Lemma 1
• proof
• Corollary 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• Lemma 2