On degree sequences of undirected, directed, and bidirected graphs

TL;DR

This paper extends classical graph degree sequence results to bidirected graphs, characterizing net-degree sequences, operations, extremal sequences, and counting formulas, highlighting similarities and differences with undirected and directed graphs.

Contribution

It generalizes key theorems and enumeration methods from undirected and directed graphs to the broader setting of bidirected graphs, including geometric insights.

Findings

Characterization of net-degree sequences for bidirected graphs

Complete degree-preserving operations for bidirected graphs

Counting formulas for net-degree sequences

Abstract

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs: - Erd\H{o}s-Gallai-type results: characterization of net-degree sequences, - Havel-Hakimi-type results: complete sets of degree-preserving operations, - Extremal degree sequences: characterization of uniquely realizable sequences, and - Enumerative aspects: counting formulas for net-degree sequences. To underline the similarities and differences to their (un)directed counterparts, we briefly survey the undirected setting and we give a thorough account for digraphs with an emphasis on the discrete geometry of degree sequences. In particular, we determine the tight and uniquely realizable degree sequences for directed graphs.