Recognizing Chordal-Bipartite Probe Graphs

A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have been studied previously on other families of graphs, such as interval graphs and chordal graphs. In this paper, we give a characterization of chordal-bipartite probe graphs, in the case of a fixed given partition of the vertices into probes and nonprobes. Our results are obtained by solving first the more general case without assuming that N is a stable set, and then this can be applied to the more specific case. Our characterization uses an edge elimination ordering which also implies a polynomial time recognition algorithm for the class. This research was conducted in the context of a France-Israel Binational project, while the French team visited Haifa in March 2007

04221 Abstracts Collection -- Robust and Approximative Algorithms on Particular Graph Classes

From 23.05.04 to 28.05.04, the Dagstuhl Seminar 04221 ``Robust and Approximative Algorithms on Particular Graph Classes\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Probe split graphs

Graphs and Algorithm

The Sandwich Problem for Decompositions and Almost Monotone Properties

This is a post-peer-review, pre-copyedit version of an article published in Algorithmica. The final authenticated version is available online at: https://doi.org/10.1007/s00453-018-0409-6We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in C as an induced subgraph is almost monotone if C is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length ≡ j mod n is almost monotone if and only if j ≡ 2 mod n or n ≤ 2. Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.Maria Chudnovsky was supported by National Science Foundation Grant DMS-1550991 and US Army Research Office Grant W911NF-16-1-0404. Celina M. H. de Figueiredo was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq Grant 303622/2011-3

Chordal Probe Graphs (Extended Abstract)

In this paper, we introduce the class of chordal probe graphs which are a generalization of both interval probe graphs and chordal graphs. A graph G is chordal probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We show that a chordal probe graph may contain neither an odd-length chordless cycle nor the complement of a chordless cycle, and we present the complete heirarchy with separating examples for these classes. We give polynomial time recognition algorithms for the subfamily of chordal probe graphs which are also weakly chordal, first in the case of a fixed given partition of the vertices into probes and non-probes, and second in the more general case where no partition is given