The Simultaneous Representation Problem for Chordal, Comparability and Permutation Graphs
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Graph Theory, Computational Intelligence and Thought
Characterisations and linear-time recognition of probe cographs
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
Probe distance-hereditary graphs
CATS '10 Proceedings of the Sixteenth Symposium on Computing: the Australasian Theory - Volume 109
Adjacency matrices of probe interval graphs
Discrete Applied Mathematics
Recognition of probe ptolemaic graphs
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Recognition of probe distance-hereditary graphs
Discrete Applied Mathematics
Recognition of probe proper interval graphs
Discrete Applied Mathematics
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A graph $G=(V,E)$ is a chordal probe graph 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 give several characterizations of chordal probe graphs, 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. In both of these cases, our results are obtained by introducing new classes, namely, $N$-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. $N$-triangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it $N$-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is $O(|P||E|)$, thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is $O(|E|^2)$.