Tatiana Romina Hartinger

Partial Characterizations of 1-Perfectly Orientable Graphs

We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this paper, we develop several results on 1-perfectly orientable graphs. In particular, we: (i) give a characterization of 1-perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1-perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1-perfectly orientable graphs, and (iv) characterize the class of 1-perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1-perfectly orientable co-bipartite graphs coincides with the class of co-bipartite circular arc graphs.

The price of connectivity for cycle transversals

For a family of graphs F , an F -transversalźof a graph G is a subset S ź V ( G ) that intersects every subset of V ( G ) that induces a subgraph isomorphic to a graph in F . Let t F ( G ) źbe the minimum size of an F -transversalźof G , and ct F ( G ) źbe the minimum size of anź F -transversalźof G that induces a connected graph. For a class of connected graphs G , we say that the price of connectivity of F -transversalsźis multiplicative if, for all G ź G , ct F ( G ) / t F ( G ) is bounded by a constant, and additive if ct F ( G ) - t F ( G ) is bounded by a constant. The price of connectivity is identical if t F ( G ) źand ct F ( G ) źare always equal and unbounded if ct F ( G ) źcannot be bounded in terms of t F ( G ) . We study classes of graphs characterized by one forbidden induced subgraphź H and F -transversalsźwhereź F contains an infinite number of cycles and, possibly, also one or more anticycles or short paths. We determine exactly those classes of connected H -free graphs where the price of connectivity of these F -transversalsźis unbounded, multiplicative, additive, or identical. In particular, our tetrachotomies extend known results for the case when F is the family of all cycles.

Minimum connected transversals in graphs : new hardness results and tractable cases using the price of connectivity.

We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph

The Price of Connectivity for Cycle Transversals

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1-perfectly orientable graphs and graph products

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On Three Extensions of Equimatchable Graphs

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Improved Algorithms for k-Domination and Total k-Domination in Proper Interval Graphs

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On total domination in the Cartesian product of graphs

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On two extensions of equimatchable graphs

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On the structure of 1-perfectly orientable graphs.

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