# Packing tree degree sequences

## TL;DR

This paper investigates conditions under which two or more tree degree sequences can be realized as edge-disjoint trees, providing polynomial-time algorithms for certain cases and proving NP-completeness for related problems.

## Contribution

It establishes new polynomial-time results for edge-disjoint realizations of tree degree sequences under specific conditions and proves NP-completeness for a related degree matrix problem.

## Key findings

- Edge-disjoint caterpillar realizations exist if two tree degree sequences have no common leaves.
- Edge-disjoint realizations always exist if each vertex is a leaf in at least one of the trees.
- Counting realizations is feasible with efficient approximations when each vertex is a leaf in at least one tree.

## Abstract

We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge packing problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. We also consider the following variant of the degree matrix problem: given two degree sequences $D_1$ and $D_2$ such that $D_2$ is a tree degree sequence, decide if there exists edge-disjoint realizations of $D_1$ and $D_2$ where the realization of $D_2$ is not necessarily a tree. We show that this problem is already $\NP$-complete. Counting, or just estimating the number of distinct realizations of degree sequences is challenging in general. We show that efficient approximations for the number of solutions as well as an almost uniform sampler exist for two tree degree sequences if each vertex is a leaf in at least one of the trees.