On finding orientations with fewest number of vartices with small out-degree

TL;DR

This paper presents a polynomial-time algorithm for assigning orientations to the edges of an undirected graph to minimize the number of vertices with out-degree at most one, addressing a question from prior research.

Contribution

It introduces a novel polynomial-time algorithm for minimizing poor vertices in graph orientations, solving an open problem from 2014.

Findings

Algorithm efficiently minimizes poor vertices in graph orientations

Provides a polynomial-time solution to a previously open problem

Addresses a question posed by Asahiro Jansson et al. in 2014

Abstract

Given an undirected graph, each of the two end-vertices of an edge can own the edge. Call a vertex poor, if it owns at most one edge. We give a polynomial time algorithm for the problem of finding an assignment of owners to the edges which minimizes the number of poor vertices. In the terminology of graph orientation, this means finding an orientation for the edges of a graph minimizing the number of edges with out-degree at most 1, and answers a question of Asahiro Jansson, Miyano, Ono (2014).