Vol. 9, No. 3, 2020

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Weakly distinguishing graph polynomials on addable properties

Johann A. Makowsky and Vsevolod Rakita

Vol. 9 (2020), No. 3, 333–349
Abstract

A graph polynomial P is weakly distinguishing if for almost all finite graphs G there is a finite graph H that is not isomorphic to G with P(G) = P(H). It is weakly distinguishing on a graph property 𝒞 if for almost all finite graphs G 𝒞 there is H 𝒞 that is not isomorphic to G with P(G) = P(H). We give sufficient conditions on a graph property 𝒞 for the characteristic, clique, independence, matching, and domination and ξ polynomials, as well as the Tutte polynomial and its specializations, to be weakly distinguishing on 𝒞. One such condition is to be addable and small in the sense of C. McDiarmid, A. Steger and D. Welsh (2005). Another one is to be of genus at most k.

Keywords
graph polynomials, random graphs, Bollobás–Pebody–Riordan conjecture, addable graph classes
Mathematical Subject Classification 2010
Primary: 05C31
Secondary: 05C10, 05C30, 05C69, 05C80
Milestones
Received: 5 December 2019
Revised: 1 March 2020
Accepted: 19 March 2020
Published: 15 October 2020
Authors
Johann A. Makowsky
Department of Computer Science
Technion - IIT
Haifa, Israel
Vsevolod Rakita
Department of Computer Science
Technion - IIT
Haifa, Israel