The ODD EVEN DELTA problem is #P-hard

TL;DR

The paper proves that computing the difference between counts of odd and even edge-induced subgraphs of size k in a graph is #P-hard, even for restricted graph classes like 3-regular bipartite planar graphs.

Contribution

It establishes the #P-hardness of the ODD EVEN DELTA problem for specific graph classes, extending complexity results to these cases.

Findings

Computing D_k is #P-hard for general graphs.

The hardness holds even for 3-regular bipartite planar graphs.

The problem remains computationally intractable under these restrictions.

Abstract

Let G=(V,E) be a graph. Let k < |V| be an integer. Let O_k be the number of edge induced subgraphs of G having k vertices and an odd number of edges. Let E_k be the number of edge induced subgraphs of G having k vertices and an even number of edges. Let D_k = O_k - E_k. The ODD EVEN DELTA problem consists in computing D_k, given G and k. We show that such problem is #P-hard, even on 3-regular bipartite planar graphs.