Local expressions of hereditary classes

TL;DR

The paper develops a unified framework, called local expressions, to characterize hereditary graph classes via forbidden equipped graphs, unifying prior templates such as forbidden orientations, linear orderings, and circular orderings. It shows that every local expression arises from a concrete functor and corresponds to a polynomial-time certificate for membership, while connecting locality to SNP-definability in logic. A central duality is established: concrete functors between relational-structure categories correspond to quantifier-free definitions, enabling constructive representations F = sh_Δ with Δ quantifier-free. The work further introduces pseudo-local classes as SNP-definable, closure-closed images of local classes, and discusses their complexity-theoretic implications and open problems. Together, these results lay a foundation for systematic study of graph-class expressibility and certify membership efficiently through logical and categorical machinery.

Abstract

A well-established research line in structural and algorithmic graph theory is characterizing graph classes by listing their minimal obstructions. When this list is finite for some class $\mathcal C$ we obtain a polynomial-time algorithm for recognizing graphs in $\mathcal C$, and from a logic point of view, having finitely many obstructions corresponds to being definable by a universal sentence. However, in many cases we study classes with infinite sets of minimal obstructions, and this might have neither algorithmic nor logic implications for such a class. Some decades ago, Skrien (1982) and Damaschke (1990) introduced finite expressions of graph classes by means of forbidden orientations and forbidden linear orderings, and recently, similar research lines appeared in the literature, such as expressions by forbidden circular orders, by forbidden tree-layouts, and by forbidden edge-coloured graphs. In this paper, we introduce local expressions of graph classes; a general framework for characterizing graph classes by forbidden equipped graphs. In particular, it encompasses all research lines mentioned above, and we provide some new examples of such characterizations. Moreover, we see that every local expression of a class $\mathcal C$ yields a polynomial-time certification algorithm for graphs in $\mathcal C$. Finally, from a logic point of view, we show that being locally expressible corresponds to being definable in the logic SNP introduced by Feder and Vardi (1999).

Figures (20)

• Figure 1: A set $\mathcal{F}$ of $2$-edge-coloured graph such that $G$ admits an $\mathcal{F}$-free $2$-edge-colouring if and only if $G$ is a co-bipartite graph (Example \ref{['ex:co-bip']})
• Figure 2: A set $\mathcal{F}$ of $2$-arc-coloured tournaments that such that a graph $G$ admits an orientation of edges and non-edges that avoids $\mathcal{F}$ if and only if it is a proper circular-arc co-bipartite graph --- orientations of edges are represented by blue arcs, and orientations of non-edges are represented by red dashed arcs.
• Figure 3: A set of structures that characterizes $3$-colourable graphs by means of forbidden linearly ordered oriented graphs. In each case the linear ordering is $v_1\le v_2\le v_3$.
• Figure 4: A set of structures that characterizes $3$-colourable graphs by means of forbidden linearly ordered $2$-edge-coloured graphs. In each case the linear ordering is $v_1\le v_2\le v_3$.
• Figure 5: A set $\mathcal{F}$ of circularly ordered oriented graphs, such that a graph $G$ admits a circular ordering together with an orientation that avoids $\mathcal{F}$ if and only if $G$ is a circularly ordered oriented graph. Here, the circular ordering is the circular ordering defined by the clock-wise motion around the dotted circumference.

Theorems & Definitions (99)

• Proposition 2.4
• proof
• Corollary 2.5
• Lemma 2.6
• Theorem 2.7
• proof
• Proposition 2.8
• proof
• Lemma 2.9
• proof