Properties of graphs specified by a regular language

Acta Informatica (2022) 59:357–385 https://doi.org/10.1007/s00236-022-00427-z ORIGINAL ARTICLE Properties of graphs specified by a regular language Volker Diekert1 · Henning Fernau2 · Petra Wolf2 Received: 12 October 2021 / Accepted: 17 June 2022 / Published online: 12 August 2022 © The Author(s) 2022 Abstract Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property Φ. What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying Φ in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet Σ, and we define a regular set G ⊆ Σ ∗ such that every nonempty word w ∈ G defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over Σ. Then, we ask whether the automaton A specifies some graph satisfying a certain property Φ. Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph F ∞ and therefore the family of finite subgraphs of F ∞ where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece. Research supported by DFG project FE 560/9-1. Preamble: We dedicate the paper to Klaus–Jörn Lange on the occasion of his 70th birthday. The conference abstract of the present paper appeared in [80]. Here, we give full proofs and we correct some mistakes. B Volker Diekert Henning Fernau Petra Wolf 1 Formal Methods in Informatics, Universität Stuttgart, Stuttgart, Germany 2 FB 4 – Informatikwissenschaften, Universität Trier, Trier, Germany 123 358 V. Diekert et al. 1 Introduction The paper is about families of finite graphs specified by regular languages, and their properties. When dealing with algorithms, a graph is often specified by its adjacency matrix or by its induced edge-list together with the number of isolated vertices, if there are any. In either representation, a graph comes with a linear order on the vertices and the edges are directed. Moreover, an adjacency matrix ignores multiple edges, but self-loops may occur. We follow these conventions in our paper. We encode1 a finite graph G = (V , E) as a word over the binary alphabet Σ = {a, b} as follows: The ith vertex u i of a graph is encoded by abi a and the  + edge (u i , u j ) is encoded by abi aaab j a. Thus, every word w in G = ab+ a ∪ ab+ aaab+ a represents in a natural way a unique graph ρ(w), because ab+ a ∪ ab+ aaab+ a is a regular code. Namely, ρ(w) is the graph consisting of all vertices and edges whose encodings appear as a factor in w ∈ G. Notice that it does not matter if a factor appears once or many times. Given a finite graph G = (V , E) with any linear order on the vertices, we obtain a code word γ (G) in G as follows. We write V as {1, . . . , |V |} using the bijection induced by the linear order on V , and then we write the edges and the isolated vertices in the order which yields the short-lex normal form of G in ρ −1 (G). This means that first, all edges are listed and then, possible isolated vertices follow. We are interested in abstract graphs, only. Thus, isomorphic graphs are treated as equal. Therefore, several γ (G)’s are possible, depending on the linear orders on V , but even then, the short-lex normal form would give a unique syntactic representation of G if necessary. We cannot avoid that every nonempty graph G has infinitely many representations w ∈ G such that ρ(w) = G. For example, the one-point graph ({}, ∅) is represented by all words in the regular set L i = (abi a)+ as soon as i ≥ 1, i.e., for all w ∈ L i we have ρ(w) = G. Given any L ⊆ G, it defines a set of graphs ρ(L). The main interest is when ρ(L) is infinite but L ⊆ G is regular. The aim is to “understand” the infinite set of graphs in ρ(L). It is far from obvious that this is possible. If L is finite, then ρ(L) is finite, too. But the converse is false. As we will see, if L is regular, then we can decide finiteness of ρ(L); and if ρ(L) is finite, then we can compute all its graphs. However, if ρ(L) is infinite, then a global understanding of ρ(L) is, a priori, not easy. 1.1 A sketch of our approach and our results Let us try to give a high-level explanation of the underlying geometric idea how to approach the family of graphs ρ(L). Remotely, it is like understanding the geometry of a topological manifold using the fact that it locally resembles an Euclidean space. For example, it is possible to realize a torus (which is a compact two-dimensional surface) as a unit square where opposite edges are identified. Every point has on open neighborhood which looks like R2 and from that one easily derive that the so-called fundamental group (which is a global property) is the group Z × Z. Therefore, we cannot transform a torus neither into a sphere nor into a soup tureen with two or more handles. In our case, we deal with purely combinatorial objects. Nevertheless, we wish to understand the set of graphs ρ(L) by constructing a finite subset of graphs together with an “open neighborhood” around these graphs such that ρ(L) is covered by that construction. Thus, if we want to check whether a certain property Φ is satisfied by some graph in ρ(L), then it is enough that we are able to check that locally. The key idea is to cut first L into pieces using the algebraic property that a regular language L has a finite syntactic monoid M L . Hence, 1 We briefly discuss our encoding of graphs as words (and some related work) in Sect. 1.2. 123 Properties of graphs specified by a regular language 359 L is a finite union of congruence classes; and we obtain an important saturation property: Whenever w ∈ L, we define the set of words [w] to be all words in the same congruence class of w. So, ρ([w]) plays the role of an open neighborhood around the graph ρ(w). Inside each ρ([w]), we define finitely many “smallest” graphs. Thereby, we find a finite set of finite graphs F such that the collection of these finitely many graphs still has the entire information about ρ(L). In order to reveal that information, we construct for each F ∈ F a (possibl (...truncated)