Unary automatic graphs: an algorithmic perspective

Abstract

1This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

Analysing Complexity in Classes of Unary Automatic Structures

“…In particular, when the automaton is fixed, deciding if there is a path from vertex x to vertex y takes linear time in x and y. Furthermore, [7] showed that connectedness of unary automatic graphs of finite degrees is decidable in O(n 3 ) time and the following result which we will use later.…”

“…Therefore, the membership problem is decidable in linear time. In [7], Khoussainov, Liu, and Minnes investigated a range of algorithmic properties of unary automatic graphs of finite degree. For example, they showed that the reachability problem for unary automatic graphs of finite degree can be decided in polynomial time in the sizes of the input vertices and the automaton.…”

“…However, [7] left open the decidability of the isomorphism problem. We now settle this question and provide an algorithm deciding the isomorphism problem for unary automatic graphs of finite degree.…”

“…The case of directed graphs can be treated in a similar manner. We use the following characterization from [7]. Given a finite graph F = (V F ; E F ) and a map σ : V F → P(V F ), we define F σ ω to be the disjoint union of infinitely many copies of F with added edges: there is an edge between x ∈ F i and y ∈ F i+1 if and only if y ∈ σ(x).…”

Deciding the isomorphism problem in classes of unary automatic structures

“…Deciding these questions by a translation of monadic second-order formulae yields very slow algorithms (non-elementary complexity). Khoussainov, Liu and Minnes [11] exploited structural properties of unary automatic graphs with finite degree to solve these questions in polynomial-time.…”