13 research outputs found
Deciding Circular-Arc Graph Isomorphism in Parameterized Logspace
We compute a canonical circular-arc representation for a given circular-arc
(CA) graph which implies solving the isomorphism and recognition problem for
this class. To accomplish this we split the class of CA graphs into uniform and
non-uniform ones and employ a generalized version of the argument given by
K\"obler et al (2013) that has been used to show that the subclass of Helly CA
graphs can be canonized in logspace. For uniform CA graphs our approach works
in logspace and in addition to that Helly CA graphs are a strict subset of
uniform CA graphs. Thus our result is a generalization of the canonization
result for Helly CA graphs. In the non-uniform case a specific set of ambiguous
vertices arises. By choosing the parameter to be the cardinality of this set
the obstacle can be solved by brute force. This leads to an O(k + log n) space
algorithm to compute a canonical representation for non-uniform and therefore
all CA graphs.Comment: 14 pages, 3 figure
Interval Routing Schemes for Circular-Arc Graphs
Interval routing is a space efficient method to realize a distributed routing
function. In this paper we show that every circular-arc graph allows a shortest
path strict 2-interval routing scheme, i.e., by introducing a global order on
the vertices and assigning at most two (strict) intervals in this order to the
ends of every edge allows to depict a routing function that implies exclusively
shortest paths. Since circular-arc graphs do not allow shortest path 1-interval
routing schemes in general, the result implies that the class of circular-arc
graphs has strict compactness 2, which was a hitherto open question.
Additionally, we show that the constructed 2-interval routing scheme is a
1-interval routing scheme with at most one additional interval assigned at each
vertex and we an outline algorithm to calculate the routing scheme for
circular-arc graphs in O(n^2) time, where n is the number of vertices.Comment: 17 pages, to appear in "International Journal of Foundations of
Computer Science
Automorphism Groups of Geometrically Represented Graphs
Interval graphs are intersection graphs of closed intervals and circle graphs are intersection graphs of chords of a circle. We study automorphism groups of these graphs. We show that interval graphs have the same automorphism groups as trees, and circle graphs have the same
as pseudoforests, which are graphs with at most one cycle in every connected component.
Our technique determines automorphism groups for classes with a
strong structure of all geometric representations, and it can be applied to other graph classes. Our results imply polynomial-time algorithms for computing automorphism groups in term of group products
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four