On the decidability of integer subgraph problems on context-free graph languages

Abstract

We show the decidability of integer subgraph problems (ISPs) on context-free sets of graphs L(Γ) defined by hyperedge replacement systems (HRSs) Γ. Additionally, we give a very general characterization of ISPs to be decidable on a set L(Γ). An ISP ∏ consists of a property s _∏ and a mapping f _∏ . If J is a subgraph of a graph G, then s _∏ (G, J) is true or false, and f _∏ (G, J) is an integer. We show the decidability of the following problem: Let Π₁,...,Π _n be n ISPs that fulfill our characterization and let C be a set of conditions (i, o, j) that specify two ISPs Π _i and Π _j and a compare symbol o ∈ {=, ≠, <, ≤, >, ≥}. Given a context-free set of graphs L defined by a HRS, is there a graph G ∈ L that has n subgraphs J ₁,...,J _n such that $s_{\prod _i }$(G, J) holds true for i = 1,...,n and $s_{\prod _i }$(G, J _i ) o $s_{\prod _j }$(G, J _j ) for each condition (i, o, j)?