The PIGs Full Monty - A Floor Show of Minimal Separators

@inproceedings{Chang2005ThePF,
  title={The PIGs Full Monty - A Floor Show of Minimal Separators},
  author={Gerard J. Chang and Ton Kloks and Jiping Liu and Sheng-Lung Peng},
  booktitle={STACS},
  year={2005}
}
Given a class of graphs $\mathcal{G}$, a graph G is a probe graph of $\mathcal{G}$ if its vertices can be partitioned into two sets ℙ (the probes) and ℕ (non–probes), where ℕ is an independent set, such that G can be embedded into a graph of $\mathcal{G}$ by adding edges between certain vertices of ℕ. We show that the recognition problem of probe interval graphs, i.e., probe graphs of the class of interval graphs, is in P. 
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29 Citations
Background Citations
13
Methods Citations
4

29 Citations

Citation Type

Citation Type

Probe Ptolemaic Graphs
TLDR
This paper studies the probe graphs of ptolemaic graphs when the partition of vertices is unknown and shows that there exists a polynomial-time recognition algorithm for probe ptolemic graphs.
On Probe Permutation Graphs
TLDR
It is shown that there are at most O(n4) minimal separators for a probe permutation graph and there exist polynomial-time algorithms solving treewidth and minimum fill-in problems for probe permutations graphs.
On the Recognition of Probe Graphs of Some Self-Complementary Classes of Perfect Graphs
TLDR
It is shown that there are polynomial-time recognition algorithms for probe cographs, probe P4-reducible graphs, probeP4-sparse graphs, and probe splitgraphs.
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TLDR
The first polynomial-time algorithm for recognizing partitioned probe distance-hereditary graphs is given and it is shown that the recognition of both partitioned and unpartitioned probe cographs can be done in ${O}(\mathfrak\it{n}^2)$ time.
Recognizing Chordal Probe Graphs and Cycle-Bicolorable Graphs
TLDR
It is proved that any cycle-bicoloring of a graph renders it $N-triangulatable, thus also providing an interesting tractable subcase of the chordal graph sandwich problem.
Recognition of Probe Ptolemaic Graphs - (Extended Abstract)
TLDR
This work proposed an O(nm)- time algorithm to recognize probe ptolemaic graphs where n and m are the numbers of vertices and edges of the input graph respectively.
Partitioned probe comparability graphs
On probe 2-clique graphs and probe diamond-free graphs
Given a class G of graphs, probe G graphs are defined as follows. A graph G is probe G if there exists a partition of its vertices into a set of probe vertices and a stable set of nonprobe vertices…
Probe Distance-Hereditary Graphs
TLDR
This paper gives an O(nm)-time algorithm for recognizing probe graphs of distance-hereditary graphs, a type of graph that is distance hereditary and has been studied by many researchers.
Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem
TLDR
This work shows that any hereditary class of graphs of bounded clique cover number that excludes some complete prism is tame, and develops a dichotomy theorem separating tame from non-tame graph classes within the family of graph classes defined by sets of forbidden induced subgraphs with at most four vertices.
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References

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