Spanning Trees with Many Leaves in Regular Bipartite Graphs

Abstract

Given a d-regular bipartite graph G _d , whose nodes are divided in black nodes and white nodes according to the partition, we consider the problem of computing the spanning tree of G _d with the maximum number of black leaves. We prove that the problem is NP hard for any fixed d ≥ 4 and we present a simple greedy algorithm that gives a constant approximation ratio for the problem. More precisely our algorithm can be used to get in linear time an approximation ratio of 2 − 2/(d − 1)² for d ≥ 4. When applied to cubic bipartite graphs the algorithm only achieves a 2-approximation ratio. Hence we introduce a local optimization step that allows us to improve the approximation ratio for cubic bipartite graphs to 1.5.

Focusing on structural properties, the analysis of our algorithm proves a lower bound on l _B (n,d), i.e., the minimum m such that every G _d with n black nodes has a spanning tree with at least m black leaves. In particular, for d = 3 we prove that l _B (n,3) is exactly $\left\lceil\frac{n}{3}\right\rceil +1$.