Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

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

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

Graph Isomorphism for unit square graphs

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.Comment: 31 pages, 6 figure

Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs