Pr\"{u}fer codes on vertex-colored rooted trees
R. W. R. Darling, Grant Fickes

TL;DR
This paper extends Pr"{u}fer codes to vertex-colored rooted trees, enabling encoding, comparison, and analysis of such trees without unique labeling, with applications to freight network decomposition.
Contribution
It introduces a canonical labeling method and a vertex-colored Pr"{u}fer code (VCPC) that captures isomorphism and subtree relations for vertex-colored rooted trees.
Findings
VCPC characterizes tree isomorphism.
VCPC encodes subtree relationships.
VCPC identifies minors of vertex-colored trees.
Abstract
Pr\"{u}fer codes provide an encoding scheme for representing a vertex-labeled tree on vertices with a string of length . Indeed, two labeled trees are isomorphic if and only if their Pr\"{u}fer codes are identical, and this supplies a proof of Cayley's Theorem. Motivated by a graph decomposition of freight networks into a corpus of vertex-colored rooted trees, we extend the notion of Pr\"{u}fer codes to that setting, i.e., trees without a unique labeling, by defining a canonical label for a vertex-colored rooted tree and incorporating vertex colors into our variation of the Pr\"{u}fer code. Given a pair of trees, we prove properties of the vertex-colored Pr\"{u}fer code (abbreviated VCPC) equivalent to (1) isomorphism between a pair of vertex-colored rooted trees, (2) the subtree relationship between vertex-colored rooted trees, and (3) when one vertex-colored rooted tree is…
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Taxonomy
TopicsAdvanced Graph Theory Research · Error Correcting Code Techniques · Interconnection Networks and Systems
