Enumerating Labeled Graphs that Realize a Fixed Degree Sequence
Atabey Kaygun

TL;DR
This paper presents a recurrence relation and recursive algorithm for counting labeled graphs that realize a fixed degree sequence, improving efficiency especially for regular graphs compared to generating function methods.
Contribution
It introduces a simple recurrence relation and an efficient recursive algorithm for exact enumeration of graphs with a given degree sequence, including regular graphs.
Findings
Exact count of labeled graphs for a fixed degree sequence via recurrence
Recursive algorithm with improved complexity for regular graphs
Better efficiency than generating function methods for certain cases
Abstract
A finite non-increasing sequence of positive integers is called a degree sequence if there is a graph with and for . In that case we say that the graph realizes the degree sequence . We show that the exact number of labeled graphs that realize a fixed degree sequence satisfies a simple recurrence relation. Using this relation, we then obtain a recursive algorithm for the exact count. We also show that in the case of regular graphs the complexity of our algorithm is better than the complexity of the same enumeration that uses generating functions.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Algorithms and Data Compression
