On friendliness between trees

TL;DR

This paper explores the concept of 'friendliness' between trees and path graphs, providing new conditions and criteria for when a tree is considered friendly to a path graph, with implications for polyhedral intersection problems.

Contribution

It introduces new sufficient conditions and criteria for determining when a tree is friendly to a path graph, advancing understanding of tree-path graph relationships.

Findings

Trees with a path containing all vertices of degree > 2 are friendly to a path graph

New sufficient conditions for tree friendliness to a path graph are established

A criterion for friendliness between trees with diameter 3 is proved

Abstract

The notion of friendliness between trees first appeared in solution of Lando's problem on intersection of polyhedra in 3-space. A tree is friendly to a path graph if edges of the tree can be numbered so that for each k,s the path between the edges k and k+1 contains either both or none of the edges k+2s,k+2s+1. Theorem. If a tree contains a path containing all vertices of degree greater than 2, then the tree is friendly to a path graph. We also prove another sufficient condition for friendliness to a path graph and a criterion for friendliness between trees, one of which has diameter 3.