Which graph motif parameters count?
Markus Bl\"aser, Radu Curticapean, Julian D\"orfler, Christian Ikenmeyer

TL;DR
This paper characterizes when graph motif parameters, which count induced subgraph copies, have a combinatorial interpretation, especially focusing on the sign of coefficients and extending to relational structures and categories.
Contribution
It provides a complete characterization of graph motif parameters with combinatorial meaning, including negative coefficients, and extends the framework to relational structures and categorical motifs.
Findings
Only positive integer coefficients preserve combinatorial interpretation.
Negative coefficients make evaluation impossible in an oracle #P setting.
General dichotomy theorem for motif parameters over categories and vector spaces.
Abstract
For a fixed graph H, the function #IndSub(H,*) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #IndSub(H,*) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative. Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is…
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Taxonomy
TopicsGraph Theory and Algorithms
