Properties of Graphs Specified by a Regular Language

Abstract

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $\varPhi$. 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 exists one graph satisfying $\varPhi$? We approach this question by using formal languages for specifying families of graphs. In particular, we show that certain graph properties can be decided by studying the syntactic monoid of the specification language.

P. Wolf—Research supported by DFG project FE 560/9-1.

1 Introduction

When dealing with algorithms on graphs, a graph is often specified by its adjacency matrix, i.e., a graph comes with a linear order on the vertices, and there are no multiple edges. We follow these conventions in our paper. Moreover, we represent graphs by words from a regular set $\mathbb {G}$ over the binary alphabet $\varSigma =\{a,b\}$. Given $w\in \mathbb {G}$, we denote by $\rho (w)$ the corresponding graph. Hence, every subset $L\subseteq \mathbb {G}$ defines a family of graphs $\rho (L)$. Although our results go beyond regular sets L, the focus and the motivation comes from a situation when L is regular. A typical question could be if some (or all) graphs in $\rho (L)$ satisfy a graph property $\varPhi$. For example: “are there some planar graphs in $\rho (L)$?” Solving this type of decision problems was the motivation to study regular realizability problems in [1, 13] and, independently, calling them $int_{\mathrm {Reg}}$-problems (intersection non-emptiness with regular languages) in [6, 14, 15].

Typical graph properties ignore the linear vertex orders and the direction of edges. For example, consider the property that the number of vertices is even. The linear order helps describe this property in Monadic Second-Order logic, MSO for short. As we will see, we encounter only four different classes $\mathcal {C}_1\subset \cdots \subset \mathcal {C}_4$ of graphs $\rho (L)$.

• 1. $\rho (L)\in \mathcal {C}_1$ if and only if the set $\rho (L)$ is finite.

• 2. $\rho (L)\in \mathcal {C}_2$ implies that $\rho (L)$ has bounded tree-width.

• 3. $\rho (L)\in \mathcal {C}_3$ implies that every connected finite bipartite graph appears as a connected component of some $G\in \rho (L)$.

• 4. $\rho (L) \in \mathcal {C}_4$ implies that every connected finite graph appears as a connected component of some $G\in \rho (L)$.

Moreover, if L is regular, then we can compute the smallest $\ell$ such that $\rho (L)\in \mathcal {C}_\ell$. We use a straightforward encoding of vertices and edges: the i-th vertex $u_i$ of a graph is encoded by $ab^ia$ and the edge $(u_i,u_j)$ is encoded by $ab^ia a ab^ja$. Since the syntactic monoid of a regular language is finite, we find some $t,p\in \mathbb {N}$ with $p\ge 1$, threshold and period, such that for every $n\in \mathbb {N}$ there is some $c\le t+p-1$ with $b^c \equiv _L b^n$ where $\equiv _L$ denotes the syntactic equivalence. The threshold t tells us that $b^c \equiv _L b^n$ implies $n=c$ for all $0\le c < t$ and $b^n \equiv _L b^{n+p}\iff n\ge t$. This is the key observation when proving that we have no more than these four classes above. If $L\subseteq \mathbb {G}$ is not regular, then the syntactic monoid $M_L$ is infinite. We find interesting examples where $M_L$ satisfies the Burnside condition that all cyclic submonoids of $M_L$ are finite. If so, then there exist $t,p\in \mathbb {N}$ with $p\ge 1$ such that the syntactic properties stated above hold for the powers of the letter b. In this case, we say that L satisfies the (b, t, p)-torsion property. This is a strong restriction, as Theorem 1 shows that for every subset $L\subseteq \mathbb {G}$ satisfying the (b, t, p)-torsion property, there exists a regular set $R\subseteq \mathbb {G}$ such that $\rho (L)=\rho (R)$. This is quite an amazing result. Its proof relies on the fact that $\rho (L)$ is determined once we know the Parikh-image $\pi _C(\mathrm {rf}(L)) \subseteq \mathbb {N}^C$, where for $w\in \mathbb {G}$, the reduced form $\mathrm {rf}(w)$ is obtained by replacing every $b^n$ by $b^c$, where c is the smallest $0\le c \le t+p-1$ such that $b^c \equiv _L b^n$. Hence, for deciding whether some graph $G\in \rho (L)$ satisfies a property, we can assume that L is regular. We are interested in decidable properties $\varPhi$, only. Thus, we assume that the set is decidable.

First consider that $\rho (L)$ is finite. Then, we can compute all graphs in $\rho (L)$ and we can output all $G\in \rho (L)$ satisfying $\varPhi$. Finiteness of $\rho (L)$ is actually quite interesting and important. It is a case where a representation of L by a DFA or a regular expression can be used for data compression. The minimal size of a regular expression (or the size of a DFA) for L is never worse than listing all graphs in $\rho (L)$, but it might be exponentially better. For a concrete and illustrative case, we refer to Example 2 for a succinct representation of all so-called crowns with at most n cusps. The compression rate becomes even better if we use a context-free grammar which produces a finite set L of words in $\varSigma ^*$, only.

The second class $\mathcal {C}_2$ implies that $\rho (L)$ has bounded tree-width. In this case, by [2, 3, 11] we know that given any property $\varPhi$ which is expressible in MSO, it is decidable whether there is a graph in $\rho (L)$ satisfying $\varPhi$. For languages $L\subseteq \mathbb {G}$ satisfying the $(b,t,p)$-torsion property, we understand when $\rho (L)$ has finite tree-width. Hence, we have Theorem 2: The satisfiability problem for MSO-sentences is decidable for language in the second class. For the other two classes, the picture changes drastically: the First-Order theory (FO for short) becomes undecidable [12]. Conversely, we are not aware of any “natural” graph property $\varPhi$ where the satisfiability problem for $\varPhi$ is not trivial for $\mathcal {C}_3$ and $\mathcal {C}_4$.

2 Notation and Preliminaries

We let $\mathbb {N}= \{0,1, 2, \ldots \}$ be the set of natural numbers and $\mathbb {N}_{\infty }= \mathbb {N}\cup \{\infty \}$. Throughout, if S is a set, then we identify a singleton set $\{x\}\subseteq S$ with the element $x\in S$. The power set of S is identified with $2^S$ (via characteristic functions). If $E\subseteq X\times Y$ is a relation, then ${E}^{-1}$ denotes its inverse relation . By $\mathrm {id}_X$, we mean the identity relation . Recall that $Y^X$ denotes the set of mappings from a set X to a set Y. If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are mappings, then $gf:X\rightarrow Z$ denotes the mapping defined by $gf(x)=g(f(x))$. If convenient, we abbreviate f([x]) as f[x]. Throughout, $\varGamma$ denotes a finite alphabet and we fix $\varSigma =\{a,b\}$ with $a\ne b$. Each alphabet is equipped with a linear order on its letters. For $\varSigma$, we let $a<b$. The linear order on $\varGamma$ induces the short-lex linear order $\mathrel {\le _\mathrm {slex}}$ on $\varGamma ^*$. That is, for $u,v\in \varGamma ^*$, we let $u \mathrel {\le _\mathrm {slex}}v$ if either $\left| \mathinner {u}\right| <\left| \mathinner {v}\right|$ or $\left| \mathinner {u}\right| =\left| \mathinner {v}\right|$, $u=pcu'$, and $v=pdv'$ where $c,d\in \varGamma$ with $c< d$. Here, $\left| \mathinner {u}\right|$ denotes the length of u. Similarly, $|u|_a$ counts the number of occurrences of letter a in u. A language $L\subseteq \varGamma ^*$ is a code if $c_1\cdots c_m = d_1\cdots d_n \in \varGamma ^*$ with $c_i,d_j\in L$ implies $m=n$ and $c_i=d_i$ for all $1\le i \le m$. If M is a monoid, then $u\le v$ means $v\in M uM$, i.e., u is a factor of v. This notation is used for the monoids $\varGamma ^*$ and $\mathbb {N}^\varGamma$. Elements in $\mathbb {N}^\varGamma$ are called vectors. For $w\in \varGamma ^*$, $\overleftarrow{w}$ is the reversal of w.

Every subset $R\subseteq \varGamma ^*$ has a syntactic monoid $M=M_R$, see, e.g., [5]. The elements of $M_R$ are the congruence classes w.r.t.the syntactic congruence $\equiv _R$. If R is regular, then $M_R$ is finite. Monoids with a single generator are called cyclic. Every finite cyclic monoid M is defined by two numbers $t,p\in \mathbb {N}$ with $p\ge 1$ such that M is isomorphic to the quotient monoid $C_{t,p}$ of $(\mathbb {N},+,0)$ with the defining relation $t=t+p$. Hence, the carrier set of $C_{t,p}$ equals $\{0,1,\dots ,t+p-1\}$. If $t=0$ and $p=1$, then $C_{t,p}$ is trivial.

Parikh-Images. If $v,w\in \varGamma ^*$, then $|w|_v$ denotes the number how often v appears as a factor in w, i.e., . If $V\subseteq \varGamma ^*$, then the Parikh-mapping w.r.t.V is defined by $\pi _V:\varGamma ^*\rightarrow \mathbb {N}^{V}$, mapping a word w to its Parikh-vector $(|w|_v)_{v\in V}\in \mathbb {N}^{V}$. The classical case is $V=\varGamma$; then the Parikh-vector becomes $(|w|_a)_{a\in \varGamma }$ and the Parikh-mapping is the canonical homomorphism from the free monoid $\varGamma ^*$ to the free commutative monoid $\mathbb {N}^\varGamma$.

A subset $S\subseteq \mathbb {N}^\varGamma$ is called positively downward-closed if, for all $v\in S$, (a) $v(z)\ge 1$ for all $z\in \varGamma$, and (b) $u\le v$ and $u(z)\ge 1$ for all $z\in \varGamma$ imply $u\in S$. The complement of a positively downward-closed set $S\subseteq \mathbb {N}^\varGamma$ is upward-closed, i.e., $u\ge v$ and $v\in S$ imply $u\in S$. An upward-closed set S is determined by its set $M_S$ of minimal elements. By Dickson’s Lemma, for every upward-closed subset $S\subseteq \mathbb {N}^\varGamma$, $M_S$ is finite. Hence, every upward-closed subset is semi-linear. As the set of all semi-linear sets in $\mathbb {N}^\varGamma$ is closed under Boolean operations, every positively downward-closed set $S\subseteq \mathbb {N}^\varGamma$ is also semi-linear, a key for Theorem 1.

Retractions and retracts. Let $\rho :X\rightarrow Y$ and $\gamma :Y\rightarrow X$ be mappings between sets. Then, $\rho$ is called a retraction and Y is called a retract of X with section $\gamma$ if $\rho (\gamma (y))=y$ for all $y\in Y$. Then, ${\rho }^{-1}(y)$ is the fiber of $y\in Y$. If $\rho :X\rightarrow Y$ is a homomorphism of groups X and Y and $H=\ker (\rho )$ is the kernel, then $\rho$ is a retraction if and only if X is a semi-direct product of H by Y.

3 Graphs

All graphs are assumed to be (at most) countable, given as a pair $G=(V,E)$ where $E\subseteq V\times V$. An undirected graph is the special case where $E={E}^{-1}$. If $G=(V,E)$ is a directed graph, then G also defines the undirected graph $(V,E\cup {E}^{-1})$; and it defines the undirected graph without self-loops $(V,(E\cup {E}^{-1})\setminus \mathrm {id}_V)$. A graph without isolated vertices is called an edge-graph; hence, specifying the edge set suffices. If $G'=(V',E')$ and $G=(V,E)$ are graphs such that $V'\subseteq V$ and $E'\subseteq E$, then $G'$ is a subgraph of graph G and we denote this fact by $G'\le G$. If $U\subseteq V$ is any subset, then $G[U] = (U,E\cap U\times U)$ denotes the induced subgraph of U in G. A graph morphism $\varphi : (V',E')\rightarrow (V,E)$ is given by a mapping $\varphi : V'\rightarrow V$ such that $(u,v)\in E'$ implies $(\varphi (u),\varphi (v))\in E$. If $(V',E')$ and (V, E) are undirected graphs without self-loops, then $\varphi : (V',E')\rightarrow (V,E)$ is a graph morphism when $(\varphi (u),\varphi (v))\in E\cup \mathrm {id}_V$. If $\varphi$ is surjective on vertices and edges, i.e., $\varphi (V')=V$ and $\varphi (E')=E$, $\varphi$ is a projection. We consider graphs up to isomorphism, only. Hence, writing $G=G'$ means that G and $G'$ are isomorphic. A graph $F=(V,E)$ is a retract of a graph $F'=(V',E')$ if there are morphism s $\varphi : F'\rightarrow F$ and $\gamma : F\rightarrow F'$ where $\varphi \gamma$ is the identity on vertices and edges of (V, E), i.e., F appears in $F'$ as the induced subgraph $F'[\gamma (V)]$.

In our paper, every word $w\in \varSigma ^*$ represents a directed finite graph $\rho (w)=(V(w),E(w))$ with a linear order on vertices as follows.

The empty word represents the empty graph: there are no vertices and no edges. We extend $\rho$ to $2^{\varSigma ^*}$ by $\rho (L)=\{\rho (w)\mid w\in L\}$. Vice versa, if $G=(V,E)$ denotes a finite graph with a linear order on its vertices, then, for $1\le i,j\in \mathbb {N}$, the i-th vertex is represented by the factor $ab^ia$, and an edge from the i-th vertex to the j-th vertex is represented by the factor $ab^iaaab^ja$. Thus, vertices are encoded by elements in the set and edges are encoded by elements is the set . Note that $\mathbb {V}\cap \mathbb {E}=\emptyset$ and $\mathbb {V}\cup \mathbb {E}$ is an infinite regular code. Using these conventions, the regular set $\mathbb {G}=(\mathbb {V}\cup \mathbb {E})^*$ as well as its subset $\mathbb {E}^*\mathbb {V}^*$ represents all finite graphs. The set $\mathbb {E}^*$ represents all edge-graphs, i.e., graphs without isolated vertices. Every nonempty finite graph has infinitely many representations. For example, there are uncountably many subsets $L\subseteq (aba)^+\subseteq \mathbb {V}^+$ and each $\rho (L)$ represents nothing but the one-point graph without self-loop. In order to choose a unique (and minimal) representation for a finite graph $G=(V,E)$, we choose the minimal word $\gamma (G)= u_1\cdots u_m v_1\cdots v_n\in \mathbb {G}$ in the short-lex ordering on $\varSigma ^*$ such that $\rho \gamma (G)=G$, $u_k\in \mathbb {E}$ for $1\le k \le m$ and $v_\ell \in \mathbb {V}$ for $1\le \ell \le n$. Each $u_k$ is of the form $ab^iaaab^j a$ representing an edge and each $v_\ell$ is of the form $ab^ia$ representing an isolated vertex. We call $\gamma (G)$ the short-lex representation of G. Since $\gamma (G)$ is minimal w.r.t.$\mathrel {\le _\mathrm {slex}}$, we have $m=\left| \mathinner {E}\right|$ and n is the number of isolated vertices. For a graph without isolated vertices, this means that it is given by its edge list. The set of all $\gamma \rho (\mathbb {G})$ is context-sensitive but not context-free. The uvwxy-Theorem does not hold for $\gamma \rho (\mathbb {G})$.

A subset $L\subseteq \mathbb {G}$ is viewed as a description of the set of graphs $\rho (L)$. The mapping $\rho :{\rho }^{-1}(\mathcal {L})\rightarrow \mathcal {L}$ is a retraction in the sense of Sect. 2, since $\rho \gamma (G)=G$ for any finite graph G. The main results of the paper are: (1) for $L\subseteq \mathbb {G}$ satisfying the $b$-torsion property, there is a regular language $R\subseteq \mathbb {G}$ with $\rho (L)=\rho (R)$ and (2) for a context-free language satisfying the $b$-torsion property (e.g., any regular language) $R\subseteq \mathbb {G}$, we have an effective geometric description of $\rho (R)$. The description is obtained as follows. Using the fact that R is regular, in a first step, we find effectively a semi-linear description of $\rho (R)$. In a second step, we compute a finite set of finite graphs. Each member F in that finite family is a retraction of some possibly infinite graph $F^\infty$. The description of each $G\in \rho (L)$ is given by selecting some F and the cardinality of every fiber. The precise meaning will become clear later. As a consequence of the description, we are able to show various decidability results. The following example serves as an illustration.

Example 1

In the following, we let $R\subseteq \mathbb {E}^*$ and $t,p\in \mathbb {N}$ with $p\ge 1$, $t>1$, such that $b^n\equiv _R b^{n+p}$ for all $n\ge t$. Since $t>1$, we have $[b]=\{b\}$. By a star, we denote a graph (V, E) such that there exists a vertex $z\in V$ with the property . Thus, a star has a center z and the directed edges are the outgoing rays of the star.

Furthermore, we assume that $R\subseteq (abaaa b^n(b^p)^* a)^+$ for some fixed $n\ge t$. This implies $t \le n<t+p$. Let $w\in R$. We have $w\in (abaaa b^n(b^p)^* a)^m$ for $m=|w|_a/5$, i.e., $w=(abaaab^{d_1}a)\cdots (abaaab^{d_m}a)$ where $d_i=n+k_ip$ with $k_i \in \mathbb {N}$ for $1\le i \le m$. The set can have any cardinality s in $\{1,\ldots ,m\}$. Therefore, $\rho (w)$ is a single star with at least one ray and at most m rays. If R is finite, then $\mathcal {F}=\rho (R)$ is an effective finite collection of stars with at least one ray and at most r rays where .

Claim: $\mathcal {F}$ is infinite if and only if there is some $M\ge |M_R|$ with $(abaaab^na)^{M}\in R$. The claim holds if , as in this case $\mathcal {F}$ is finite. Thus, let . Then, there is some $w\in R$ such that $abaaab^na$ appears at least $|M_R|$ times as a factor. This implies that there is some $M\ge |M_R|$ such that $(abaaab^na)^{M}\in R$. The claim follows. Moreover, if $\mathcal {F}$ is infinite, then $\mathcal {F}$ is the set of all finite stars with at least one ray.

One can show that $S=(aba a ab^2b^*a)^*(aba)$ is locally testable and therefore star-free. Hence, the set of all finite stars is specified by a star-free subset of $\varSigma ^*$.

We study properties of graphs specified by languages $L\subseteq \mathbb {G}$. If L can be arbitrary, then we can specify uncountably many families of graphs. So, we cannot expect any general decidability results. Hence, we restrict our attention to subsets $L\subseteq \mathbb {G}$ where membership for $\rho (L)$ is decidable. In fact, membership for $\rho (L)$ might be decidable although membership for L is undecidable. By Corollary 1, the following definition yields a sufficient condition for decidability of $\rho (L)$.

Definition 1

Let $b\in \varGamma$ be a letter. A subset $L\subseteq \varGamma ^*$ satisfies the (b, t, p)-torsion property if we have: $b^{t}\equiv _L b^{t+p}.$ It satisfies the b-torsion property if there are $t,p\in \mathbb {N}$ with $p\ge 1$ such that L satisfies the $(b,t,p)$-torsion property.

Every regular language $R\subseteq \varGamma ^*$ satisfies the b-torsion property because the syntactic monoid $M_R$ is finite. The language $\{wa\overleftarrow{w}\mid w\in \{aba, ab^2a\}^*\}$ is not regular, but it satisfies the b-torsion property for $t=3$ and $p=1$. The b-torsion property is exceptional if R is not regular: even deterministic linear context-free one-counter languages do not satisfy this property, in general. Consider . Clearly, $b^{k}\equiv _R b^{m}\iff k=m$.

Remark 1

Let $b\in \varGamma$ and $\left| \mathinner {\varGamma }\right| =m$. Let $L\subseteq \varGamma ^*$ with $M_L$ as its syntactic monoid. If all cyclic submonoids of $M_L$ are finite, L satisfies the b-torsion property. In the following, let $1\le p\in \mathbb {N}$. Recall that the quotient monoid defines the free Burnside group $\mathcal {B}(m,p)$. It is a group because every x has the inverse element $x^{p-1}$ as $p\ge 1$. For p large enough, Adjan showed in the 1970s that $\mathcal {B}(2,p)$ is infinite, answering a question of Burnside from 1902. A group is called p-periodic if it is the homomorphic image of some $\mathcal {B}(m,p)$.

Let $\varphi :\varGamma ^* \rightarrow G$ be a surjective homomorphism to a group G. Then, the Word Problem of G denotes the set . It is a classical fact that the syntactic monoid of $\hbox {WP}(G)$ is the group G itself. Kharlampovich constructed in [7] a periodic group B(2, p) where the Word Problem is undecidable. Since the B(2, p) is periodic, the $b$-torsion property holds trivially. Therefore, as we will see, there exists a regular subset R such that $\rho (\hbox {WP}(B(2,p)))= \rho (R)$.

For the rest of the paper, if $L\subseteq \varSigma ^*$ satisfies the $b$-torsion property, then the cyclic submonoid of $M_L$ generated by the letter b is isomorphic to $C_{t,p}$. That is, we have $t,p\in \mathbb {N}$ with $p\ge 1$, where $t+p$ is minimal such that . Moreover, we assume that L is specified such that on input $n\in \mathbb {N}$, we can compute the value $0\le c\le t+p-1$ with $b^n\equiv _L b^c$. This assumption is satisfied if L is regular and specified, say, by some NFA. For $L\subseteq \mathbb {G}$, we have $[ab^ca] = a [b^c] a$ and $[ab^caaab^da] = a [b^c] aaa[b^d]a$.

Definition 2

Let $L\subseteq \mathbb {G}$ satisfy the $(b,t,p)$-torsion property according to Definition 1. For every $[b^n]$, we define its reduced form by $\mathrm {rf}{[b^n]}= b^c$ if $[b^c] = [b^n]$ and $0\le c\le t+p-1$. Given $w\in \mathbb {G}$, we define the reduced form $\mathrm {rf}(w)$ by replacing every factor $ab^ma\le w$ by $a\, \mathrm {rf}{[b^m]}a$. The saturation $\widehat{w}$ of w is defined by replacing every factor $ab^ma\le w$ by the set $a[b^{m}]a$. Hence, $\mathrm {rf}(w)\in \widehat{w}\subseteq \mathbb {G}$.

Remark 2

Let $L\subseteq \mathbb {G}$ satisfy the $(b,t,p)$-torsion property. By possibly decreasing t and/or p, we may assume that for every $1\le c \le t+p-1$, there is some $w\in L$ such that $ab^ca\le \mathrm {rf}(w)$. Moreover, we have $[b^c] = \{b^c\}$ if and only if $c<t$.

Lemma 1

Let $L\subseteq \mathbb {G}$ satisfy the $(b,t,p)$-torsion property. Then, for every $w\in \mathbb {G}$, $w\in L \iff \widehat{w}\subseteq L \iff \mathrm {rf}(w)\in L\,$.

The $(b,t,p)$-torsion property is trivially satisfied if $L\subseteq \mathbb {G}$ is a finite set, an interesting case motivated by data compression. As mentioned in Sect. 1: if L is finite, then the minimal size of a regular expression for L is never worse than listing all graphs in $\rho (L)$, but it might be exponentially better. This type of data compression with formal language methods is also applied in practice [8, 9].

Example 2

Let $G=([n],E)$ be a connected planar graph with vertices $1,\ldots ,n$. Then, for every subset $S\subseteq \{n+1,\ldots ,2n\}$, we define a graph $G_S$ by The family might contain up to $2^{\Omega (n)}$ connected planar graphs, e.g., if G is a cycle of n nodes. If we embed G in the 2-dimensional sphere where the additional edges are spikes pointing out of the sphere, then $G_S$ can be visualized as a discrete model of a 3-dimensional “crown with at most n cusps”. One can write down a 2n-fold concatenation of finite sets describing a finite set $L_n\subseteq \mathbb {G}$ with $\rho (L_n)=\mathcal {C}_n$. The size of the corresponding regular expression is $\mathcal {O}(n^{2})$. This leads to a polynomial-size blueprint potentially producing a family of exponentially many “crowns”.

Definition 3

Let $L\subseteq \mathbb {G}$ satisfy the $(b,t,p)$-torsion property. We introduce two new finite and disjoint alphabets (depending on L)

Note that $A\subseteq BaB$. By C, we denote the union of A and B, which is also a finite alphabet with a linear order between letters given by the following definition:

$$\begin{aligned} x\le _C y\iff xy\in AB\vee (xy\in (AA \cup BB) \wedge x\mathrel {\le _\mathrm {slex}}y). \end{aligned}$$

(1)

The linear order $\le _C$ on C defines a short-lex ordering on $C^*$. Actually, C is a code. Moreover, if $uxv\in C^+$ with $x\in A$ and $u,v\in \varSigma ^*$, then $u,v\in C^*$. The analogue for $y\in B$ does not hold, in general. As C is a code, the inclusion $C\subseteq \varSigma ^*$ yields an embedding $h_C:C^+ \rightarrow \varSigma ^+$. If G is a finite graph, then the minimal element in $\mathbb {H}={h}^{-1}_C({\rho }^{-1}(G))\cap A^*B^*$ w.r.t.the short-lex ordering for words in $C^*$ is the same as the minimal element in $h_C(\mathbb {H})$ w.r.t.the ordering $a<b$. We assume henceforth that C only contains factors of words from L.

Lemma 2

Let L, C, and $\mathrm {rf}$ as in Definitions 2 and 3. Let $v\in C^*$ and $w\in L$ such that $\pi _C(v)\le \pi _C(\mathrm {rf}(w))$. If $\pi _C(v)(z)\ge 1$ for all $z\in C$, then we have $\rho (v)\in \rho (L)$.

Theorem 1

Let $L\subseteq \mathbb {G}$ be any language satisfying the $b$-torsion property. Then, there is a regular set $R\subseteq \mathbb {G}$ such that $\rho (L)= \rho (R)\,.$

Corollary 1

Let $L\subseteq \mathbb {G}$ satisfy the $b$-torsion property. Then, given a finite graph $G=(V_G,E_G)$ as an input, it is decidable whether $G\in \rho (L)$.

Corollary 2

Let $L\subseteq \mathbb {G}$ be context-free satisfying the $(b,t,p)$-torsion property. Then, we can effectively calculate a regular set $R\subseteq \mathbb {G}$ such that $\rho (R)=\rho (L)$.

Let $R\subseteq \mathbb {G}$ be regular. Then, it is well-known that there might be a much more concise representation by some context-free language $K\subseteq \mathbb {G}$ such that $\pi _C(K)=\pi _C(R)$ and hence $\rho (K)=\rho (R)$.

By Theorem 1, we know that regular languages suffice to describe all sets $\rho (L)$ where $L\subseteq \mathbb {G}$ satisfies the $b$-torsion property. Therefore, we restrict ourselves to regular languages. In the following, $R\subseteq \mathbb {G}$ denotes a regular language. Hence, we can calculate numbers $t\ge 0$ and $p\ge 1$ such that R satisfies the $(b,t,p)$-torsion property. As R is regular, the set $L= {h}^{-1}_C(R)\cap A^*B^*$ is regular; its Parikh-image $\pi _C(L)\subseteq \mathbb {N}^{C}$ is effectively semi-linear. Thus, for some finite set J:

$$\begin{aligned} \pi _C(L)=\bigcup _{j\in J} (q_j + \sum _{i\in I_j}\mathbb {N}p_i )\,, \end{aligned}$$

(2)

where $q_j, p_i \in \mathbb {N}^{C}$ are vectors. Splitting $\pi _C(L)$ into more linear sets by making the index set J larger and the sets $I_j$ smaller (if necessary), we can assume without restriction that for all $j\in J$ and $z\in C$ we have $\sum _{i\in I_j}p_i(z) \le q_j(z)$. To see this, let $1\in I_j$. Then, we have

$q_j + \sum _{i\in I_j}\mathbb {N}p_i= (q_j + \sum _{i\in I_j\setminus \{1\}}\mathbb {N}p_i ) \cup (q_j + p_1 + \sum _{i\in I_j}\mathbb {N}p_i )\,.$

Splitting L into even more but finitely many cases, we can assume without restriction (for simplifying the notation) that the set J is a singleton. Thus, $\pi _C(L)=q + \sum _{i\in I}\mathbb {N}p_i$ for some $q, p_i \in \mathbb {N}^{C}$ such that $\sum _{i\in I}p_i(z) \le q(z)$. By possibly reducing A, B, C, we may assume that $q(z)\ge 1$ for all $z\in C$ and $C=A\cup B$. In order to understand the set of graphs in $\rho (R)$, it suffices to understand the set of finite graphs defined by linear sets of the form $S=q + \sum _{i\in I}\mathbb {N}p_i\subseteq \mathbb {N}^C$, where $q(z)\ge 1$ for all $z\in C$ and $\sum _{i\in I}p_i\le q$. For that purpose, we let $r= \sum _{i\in I}p_i\le q$ and we define a function $\alpha :C\rightarrow \mathbb {N}_\infty$ as follows.

$$\begin{aligned} \alpha (z)= {\left\{ \begin{array}{ll} q(z)&{} \text {if } r(z) = 0 \wedge \exists m\in \mathbb {N}: t\le m \wedge ab^ma\le z\\ \infty &{} \text {if } r(z) \ge 1 \wedge \exists m\in \mathbb {N}: t\le m \wedge ab^ma\le z\\ 1&{} \text {otherwise. That is: } \forall m\in \mathbb {N}: ab^ma\le z \implies m<t. \end{array}\right. } \end{aligned}$$

(3)

For all $z\in C$, let $L_z\subseteq \varSigma ^*$. Then, we write $\prod _{z\in C}L_z= L_{z_1} \cdots L_{z_{\left| \mathinner {C}\right| }}\,,$ where $z_i\le z_j$ for all $i\le j$ according to the linear order defined in Eq. (1). Observe that $\prod _{z\in C}L_z$ is regular if all $L_z$ are regular. With this notation, we define:

$$\begin{aligned} R_\alpha =\prod _{z\in C}z^{\alpha (z)} \quad \text {and} \quad L_\alpha =\prod _{z\in C}[z]^{\alpha (z)} \end{aligned}$$

(4)

Notice that $L^\infty$ is just another notation for $L^+$ if L is any set of words.

Lemma 3

The sets $R_{\alpha }$, $L_{\alpha }$ of Eq. (4) are regular with $R_{\alpha }\subseteq L_{\alpha }$, $\rho (L_{\alpha }) = \rho (R)$.

Now, we define for $\alpha$ a finite family of finite graphs $\mathcal {F}_\alpha$ and then, for each $F\in \mathcal {F}_\alpha$, we define a possibly infinite graph $F^\infty$, using the notion of marked graphs.

Definition 4

For $z\in C$, let $\alpha '(z)= \alpha (z)$ if $\alpha (z)<\infty$ and $\alpha '(z)= 1$, otherwise. We let $R'_\alpha =\prod _{z\in C}z^{\alpha '(z)}$, and we define $\mathcal {F}_\alpha = \rho (R'_\alpha )$.

Since $R'_\alpha$ is a finite set of words, $\mathcal {F}_\alpha$ is a finite set of finite graphs. Now, we define the crucial notion of a marked graph, with some vertices and edges marked.

Definition 5

A marked graph is a tuple $F=(V_F,E_F,\mu )$, where $(V_F,E_F)$ is a finite graph and $\mu \subseteq V_F\cup E_F$ denotes the set of marked vertices and edges. Isolated vertices may appear, but if an isolated vertex is marked, then there is exactly one isolated vertex. We also require that whenever an edge (u, v) is marked, then at least one of its endpoints is marked, too. A marked edge-graph is a marked graph without isolated vertices.

In the following, each graph $(V_F,E_F)\in \mathcal {F}_\alpha$ as in Definition 4 defines a marked graph $F=(V_F,E_F,\mu )$ as follows, where $\mu$ denotes a marking (as in Definition 5) that we call the canonical marking. We begin by marking those vertices and edges $z\in C$ where $\alpha (z)=\infty$. In particular, if $z\in C$ is marked, then $[z]\ne \{z\}$ and [z] is an infinite set. In the second step, we mark also all vertices u which satisfy $[u]\ne \{u\}$ and which appear as an endpoint in some marked edge. Thereafter, every marked edge contains at least one marked endpoint. In the third step, if an isolated vertex is marked, then remove all isolated marked vertices $y\in B$ except one isolated vertex which is marked. In particular, after that procedure, if a marked isolated vertex y appears, then $\alpha (y)= \infty$.

Now, we switch to a more abstract viewpoint. We let $\mathcal {F}$ be any finite family of marked graphs. For each $F=(V_F,E_F,\mu )\in \mathcal {F}$, we define a possibly infinite graph $F^\infty$ where $(V_F,E_F)$ appears as an induced subgraph, and we define a family of finite graphs $\mathcal {G}_{F}$. We consider finitely many $\mathcal {F}_\alpha$, and then we study , where $F=(V_F,E_F,\mu )$ is the marked graph obtained by the canonical marking procedure above (which might have removed isolated marked vertices). For understanding $\rho (R)$, we need to describe sets $\mathcal {G}_F$ for marked graphs $F=(V_F,E_F,\mu )$. This requires to define $F^\infty$ as follows.

Definition 6

Let $F=(V_F,E_F,\mu )$ be a marked graph as in Definition 5. Then, the graph $F^\infty =(V_F^\infty ,E_F^\infty )$ is defined as follows.

with $E_F\times \{0\}=\{((u,0),(v,0))\mid (u,v)\in E_F\}$. The family $\mathcal {G}_F$ is the set of finite subgraphs of $F^\infty$ containing $(V_F\times \{0\},E_F\times \{0\})$ as an induced subgraph.

Observe that $F^\infty =F$ if and only if there is no marking, i.e., if $\mu =\emptyset$. We embed F into $F^\infty$ by a graph morphism $\gamma$ which maps each vertex $u\in V_F$ to the pair $\gamma (u) = (u,0)\in V^\infty _F$. The projection onto the first component $\varphi (u,k)=(u)$ yields a retraction for every $G\in \mathcal {G}_F$ with retract F. If no isolated vertex is marked, then $F^\infty$ has at most $|V_F|$ isolated vertices, but if there are marked vertices, then for every sufficiently large k, there is some graph in $\mathcal {G}_F$ which has exactly k isolated vertices. In order to understand the graphs in $\mathcal {G}_F$ (which is our goal), it is enough to understand the graphs G satisfying $F\le G\le F^\infty$. For $F=F^\infty$, we know everything about that set. Let us hence consider $F\ne F^\infty$. Proposition 1 shows that $\rho (R)$ is rather rich as soon as some $F\in \mathcal {F}_\alpha$ satisfies $F\ne F^\infty$. Confer the next result with the classification $\mathcal {C}_1\subset \cdots \subset \mathcal {C}_4$ of graphs from Sect. 1.

Proposition 1

Let $F=(V_F,E_F,\mu )$ be any marked graph.

• 1. If F contains a marked edge (u, v) where v is marked, then every finite star with center (u, 0) appears as an induced subgraph of some $G\in \mathcal {G}_F$.

• 2. Suppose we represent a bipartite graph as a triple (U, V, E) where $U\cap V=\emptyset$ and $E\subseteq U\times V$. Let H be any finite bipartite edge-graph. If F contains a marked edge (u, v) where u and v are marked, then a disjoint union of F and H appears in $\mathcal {G}_F$.

• 3. Let H be any finite graph. If F contains a marked self-loop (u, u), then the disjoint union of F and H belongs to $\mathcal {G}_F$.

• 4. Let $F$ be any marked graph such that one or two vertices are marked. Then, the following holds. A disjoint union of F and any non-bipartite graph appears in $\mathcal {G}_F$ if and only if there is some marked self-loop in F.

By Schützenberger’s classical theorem [10] characterizing star-freeness via finite and aperiodic syntactic monoids, this case distinction entails:

Corollary 3

Let $L\subseteq \mathbb {G}$ be any language satisfying the $b$-torsion property. If there is a star-free language R such that $\rho (L)=\rho (R)$, then there is no $F\in \mathcal {F}_\alpha$ such that a disjoint union of F and a triangle appears in $\mathcal {G}_F$.

4 Graph Properties

Throughout this section, F denotes a marked graph and $\mathcal {G}_F$ denotes the family of graphs defined in Definition 6. A graph property is a decidable subset $\varPhi \subseteq \mathbb {G}$. For a finite graph G, we write $G\models \varPhi$ if the short-lex representation $\gamma (F)$ belongs to $\varPhi$. Given a word $w\in \mathbb {G}$, we can compute $\gamma \rho (w)$. Hence, we can assume ${\rho }^{-1}(\rho (\varPhi ))= \varPhi .$ As $\rho (w)$ is realized as a graph with a natural linear order on the vertices, we have $ab^ca\le ab^da \iff c\le d$. We consider properties of undirected finite graphs, only: if $u\in \mathbb {G}$ represents the graph $\rho (u)= (V,E)$, then $\rho (u)\models \varPhi$ if and only if $(V,E\cup {E}^{-1})\models \varPhi$. We focus on the satisfiability problem $\hbox {Sat}(\mathcal {G}_F,\varPhi )$:

• Input: A marked graph F.

• Question: “$\exists G\in \mathcal {G}_F: G\models \varPhi$?”

For various well-studied graph properties, $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is decidable. For example, when $\varPhi$ states that a graph is planar, or k-colorable, etc. This follows from:

Proposition 2

Let either $\mathcal {G}_F$ be finite or $\varPhi$ be any graph property which is closed under taking induced subgraphs (or both). Then, $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is decidable.

In many cases, graph properties are expressible either in MSO or even in FO. MSO is a rich and versatile class to define graph properties¹. Since $w\in \mathbb {G}$ defines graphs with a linear order, we can express in MSO, for example, that the number of vertices is even. We use the following well-known results as black boxes. First, (Trakhtenbrot’s Theorem) [12]: given an FO-sentence $\varPhi$, it is undecidable whether there exists a graph (resp. bipartite graph) satisfying $\varPhi$. Second, given an MSO-sentence $\varPhi$ and $k\in \mathbb {N}$, it is decidable whether there exists a graph of tree-width at most k satisfying $\varPhi$, see, e.g., [2, 3, 11].

Theorem 2

Let $\varPhi$ be an MSO-sentence. Then, $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is decidable for marked graphs $F=(V_F,E_F,\mu )$ if at most one endpoint of each edge is marked.

Theorem 3

Let $\varPhi$ be an FO-sentence. Then, $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is undecidable for marked graphs $F=(V_F,E_F,\mu )$ where both endpoints of some edge are marked.

Some graph properties where the problem $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is trivially decidable are covered by the next theorem, including the problem whether $\mathcal {G}_F$ contains a non-planar graph, and various parametrized problems like: “Is there some $(V_G,E_G)\in \mathcal {G}_F$ with a clique bigger than $\sqrt{|V_G|}$?”.

Theorem 4

Let F be any marked graph and $\varPhi$ be a non-trivial graph property such that $G\models \varPhi$ if and only if there is a connected component $G'$ of G such that $G'\models \varPhi$. Then, the answer to $\hbox {Sat}(\mathcal {G}_F,\varPhi )$ is “Yes” in the following two cases. (a) The property $\varPhi$ is true for some bipartite edge-graph and there is some marked edge where both endpoints are marked. (b) There is some marked self-loop.

Example 3 lists a few graph properties which are not covered by the results above, but nevertheless the satisfiability problem is decidable.

Example 3

Let $F=(V_F,E_F,\mu )$ denote a marked graph as input. Then, the following problems are decidable. Is there some $G\in \mathcal {G}_F$ (a) with a Hamiltonian cycle (See [4]), or (b) with a perfect matching, or (c) with a dominating set of size at most $\sqrt{|V_G|}$?

Perfect Matching. Let $V_F=\{x_1,\ldots ,x_k\}$ and suppose some $G=(V_G,E_G)\in \mathcal {G}_F$ has a perfect matching. We have $V_F\subseteq V_G$. Hence, all $x_i\in V_F$ are matched by vertices $y_i\in V_G$. The induced subgraph $G[V_F\cup \{y_1,\ldots ,y_k\}]$ has a perfect matching with at most $2|V_F|$ vertices. All such small $G\in \mathcal {G}_F$ can be enumerated.

Dominating Set. If F contains no marked edge, then decide if the property holds for $(V_F,E_F)$. Otherwise, there is a marked edge $(u,v)\in E_F$. Let $V_G=(V_F\times \{0\})\cup (\{v\}\times \{1,\dots ,|V_F|^2\})$ and $E_G=E_F\cup \{((u,0),(v,i))\mid 1\le i\le |V_F|^2\}$. Then, $G=(V_G,E_G)\in \mathcal {G}_F$. Also, $V_F$ is a sufficiently small dominating set of G.

With the results above, we have a meta-theorem for graph properties $\varPhi$ with a decidable satisfiability problem, covering all cases where we have positive results.

Theorem 5

Let $r:\mathbb {N}\rightarrow \mathbb {N}$ be a non-decreasing computable function and let $\varPhi$ be a graph property such that, for each marked graph $F=(V_F,E_F,\mu )$, if some graph in $\mathcal {G}_F$ satisfies $\varPhi$, then there is a graph $G=(V,E)\in \mathcal {G}_F$ such that $G\models \varPhi$ and $|V|\le r(|V_F|)$. Then, given as input a context-free language $L\subseteq \mathbb {G}$ satisfying the $(b,t,p)$-torsion property, $\hbox {Sat}(\rho (L),\varPhi )$ is decidable.

5 Conclusion and Open Problems

The starting point of our paper was the following idea: Decide a graph property $\varPhi$ not for a single instance as in traditional algorithmic graph theory, but generalize this question to a set of graphs specified by a regular language. We chose a natural representation of graphs by words over a binary alphabet $\varSigma$, but other choices would work equally well. Next, pick your favorite graph property $\varPhi$. For example, $\varPhi$ says that the number of vertices is a prime number. The property does not look very regular, there is no way to express the property, say, in MSO. Still, given a context-free language $L\subseteq \varSigma ^*$ which satisfies the $b$-torsion property and which encodes sets of graphs, we can answer the question if there exists a graph represented by L and which satisfies $\varPhi$. This is a consequence of Theorem 5 and Bertrand’s postulate that for all $n\ge 1$, there is a prime between n and 2n.

Various problems remain open. For instance, given a graph property $\varPhi$, we can define . Suppose that ${\rho }^{-1}(\mathcal {G}(\varPhi ))$ is regular. Given a regular language $R\subseteq \varSigma ^*$, can we decide whether $\mathcal {G}(\varPhi ) \subseteq \rho (R)$? What about the equality $\mathcal {G}(\varPhi ) = \rho (R)$? We can ask the same two questions if R is context-free. Future research should address complexity issues. For example, given a typical NP-complete graph property $\varPhi$ and ask how complex it is to decide the satisfiability for $\mathcal {G}_F$ if the input is a marked graph $F$.

Note: Missing proofs can be found in [4].