The Optimization of Signed Trees

TL;DR

This paper investigates the conditions under which signed trees can realize specific signed degree sets, establishing key properties, minimal diameters, and orders for such trees based on the set characteristics.

Contribution

It characterizes when a signed degree set can be realized by a tree and determines minimal diameter and order for such trees based on the set.

Findings

A signed degree set D is realizable by a tree iff 1 or -1 is in D.

The minimal diameter of a tree realizing D is identified.

The minimal order of a tree for nonnegative D is determined.

Abstract

A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by the number of negative edges incident to $v$. Finally, we say $G$ realizes the set $D$ if: $$ D = \{sdeg(v) \text{ : } v\in V(G) \}. $$ The topic of signed degree sets and signed degree sequences has been studied from many directions. In this paper, we study properties needed for signed trees to have a given signed degree set. We start by proving that $D$ is the signed degree set of a tree if and only if $1\in D$ or $-1\in D$. Further, for every valid set $D$, we find the smallest diameter that a tree must have to realize $D$. Lastly, for valid sets $D$ with nonnegative numbers, we find the smallest order that a tree must have to realize $D$.