Graph functionality
Bogdan Alecu, Aistis Atminas, Vadim Lozin

TL;DR
This paper introduces the concept of graph functionality, a parameter that generalizes several existing graph parameters, and demonstrates its boundedness in classes of graphs with unbounded degeneracy and clique-width.
Contribution
The paper defines graph functionality and proves it generalizes parameters like degeneracy and clique-width, also showing classes with unbounded degeneracy can have bounded functionality.
Findings
Functionality generalizes degeneracy and clique-width.
Permutation, unit interval, and line graphs have bounded functionality.
Bounded functionality implies bounded VC-dimension.
Abstract
Let be a graph and its adjacency matrix. We say that a vertex is a function of vertices if there exists a Boolean function of variables such that for any vertex , . The functionality of vertex is the minimum such that is a function of vertices. The functionality of the graph is , where the maximum is taken over all induced subgraphs of . In the present paper, we show that functionality generalizes simultaneously several other graph parameters, such as degeneracy or clique-width, by proving that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalization is proper by revealing classes of graphs of unbounded degeneracy…
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Taxonomy
TopicsGraph theory and applications · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
