Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

TL;DR

This paper proves new bounds on the number of leaves in spanning trees for graphs without diamonds and blossoms, and develops an improved fixed-parameter tractable algorithm for the MAX LEAF SPANNING TREE problem.

Contribution

It generalizes previous bounds to broader classes of graphs and introduces an efficient FPT algorithm for finding spanning trees with many leaves.

Findings

Graphs without diamonds and blossoms have spanning trees with at least (n+4)/3 leaves.

The new bounds are tight and necessary for certain classes of graphs.

The FPT algorithm improves the complexity for the MAX LEAF SPANNING TREE problem.

Abstract

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3+c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for MAX LEAF SPANNING TREE.