Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace

Abstract

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property.Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.

Canonical Representations for Circular-Arc Graphs Using Flip Sets

“…This means the arc ρ(v) does not contain the point x for every vertex v ∈ V (λ). Consider the cs di cc cd ov Z 10 cd cc di cs ov Z 11 di cs cd cc ov representation ρ (X) ∈ N ((λ (X) ) (X) ) = N (λ). Then it can be checked that ρ (X) (v) contains the point x iff v is in X and therefore X is a flip set with respect to λ.…”

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter II: algorithms

“…In the full version of [21], Kobler et al ask whether it is possible to solve Rep in (deterministic) logspace. In this section we provide an affirmative answer to this question by showing that the algorithm of the previous section can be implemented so as to run in logspace.…”

“…Unfortunately, this idea is not feasible at first sight because we cannot claim L = { ∈ N | M is equivalent to a (c, , d, d s )-CA model for some c ∈ N} to be a range. For instance, C 4 11 admits a (22,9)-CA model, but it admits no (c, 10)-CA model, whatever value of c is. This is just one more example of a property that is lost when the linear structure of PIG models is replaced by the circular structure of PCA graphs as L = [ * , ∞) when M is PIG.…”

“…Moreover, Fine and Harrop [9] developed, in 1957, an effective method to transform a weak mapping of an array (i.e., a PIG model) into a uniform mapping of the same array (i.e., a PIG model of a power of a path); this algorithm is actually the first proof of Robert's PIG=UIG theorem, as far as our knowledge extends. Linear time algorithms for Rep are known since more than two decades [6,25,27] and, recently, a logspace implementation has been devised [21].The research on RepExt and BoundRep did not begin until recently and, consequently, they are not as studied as Rep. We remark that these problems are defined not only for UIG graphs, but for several graph classes with geometric representations. In the last few years, the partial representation extension and the bounded representation problems were studied for several graph classes [1-4, 16, 17, 19].…”

On the isomorphism problem for Helly circular-arc graphs

“…A few natural subclasses of CA graphs have received special attention among researchers. In particular, for proper CA graphs both the recognition and the isomorphism problems are solved in linear time, respectively, in [13] and in [4], and in logarithmic space in [10]. The latter result actually gives an logspace algorithm for canonical representation of proper CA graphs, and such an algorithm is also known for unit CA graphs [18].…”