An Optimal Realization Algorithm for Bipartite Graphs with Degrees in Prescribed Intervals

TL;DR

This paper introduces an efficient algorithm for constructing bipartite graphs with degree constraints within prescribed intervals, achieving optimal runtime and producing edge-minimal or edge-maximal graphs.

Contribution

The paper presents a new realization algorithm with linear time complexity for bipartite graphs with degree intervals, improving efficiency over previous methods.

Findings

Algorithm runs in O(|U| + |V| + |E|) time

Produces edge-minimal bipartite graphs

Can be adapted to generate edge-maximal graphs

Abstract

We consider the problem of constructing a bipartite graph whose degrees lie in prescribed intervals. Necessary and sufficient conditions for the existence of such graphs are well-known. However, existing realization algorithms suffer from large running times. In this paper, we present a realization algorithm that constructs an appropriate bipartite graph G=(U,V,E) in O(|U| + |V| + |E|) time, which is asymptotically optimal. In addition, we show that our algorithm produces edge-minimal bipartite graphs and that it can easily be modified to construct edge-maximal graphs.