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2020 International Conference on Computational Science and Computational Intelligence (CSCI), 2020 The strong perfect graph theorem is the proof of the famous Berge’s conjecture that the graph is perfect if and only if it is free of odd holes and odd anti-holes. The conjecture was settled after 40 years in 2002 by Maria Chudnovsky et. al. and the proof was published in 2006.
Maher Heal
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The Strong Perfect Graph Theorem for a Class of Partitionable Graphs
, 1984 A simple adjacency criterion is presented which, when satisfied, implies that a minimal imperfect graph is an odd hole or an odd antihole. For certain classes of graphs, including K 1,3 -free graphs, it is straightforward to validate this criterion and thus establish the Strong Perfect Graph Theorem for such graphs.
Alan Tucker +2 more
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Critical (P5,dart)-Free Graphs
International Conference on Combinatorial Optimization and Applications, 2023Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. Let $P_t$ be the path on $t$ vertices.
Wen Xia +3 more
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An optimal χ ‐bound for ( P 6 , diamond)‐free graphs
Journal of Graph Theory, 2021In this paper we show that every ( P 6 , diamond)‐free graph G satisfies χ ( G ) ≤ ω ( G ) + 3 , where χ ( G ) and ω ( G ) are the chromatic number and clique number of G , respectively.
K. Cameron +2 more
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Colouring graphs with no induced six-vertex path or diamond
International Computing and Combinatorics Conference, 2021The diamond is the graph obtained by removing an edge from the complete graph on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced subgraph isomorphic to a six-vertex path or a diamond. In this paper we show that the chromatic number
J. Goedgebeur +3 more
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Towards a constructive formalization of Perfect Graph Theorems
Indian Conference on Logic and Its Applications, 2018Interaction between clique number \(\omega (G) \) and chromatic number \(\chi (G) \) of a graph is a well studied topic in graph theory. Perfect Graph Theorems are probably the most important results in this direction. Graph G is called perfect if \(\chi
Abhishek Kr Singh, R. Natarajan
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Perfect Graph Modification Problems: An Integer Programming Approach
arXiv.orgGraph modification problems, which aim to find a small set of modifications to a graph so that it satisfies a desired property, have been studied for several special graph classes.
Burak Nur Erdem +2 more
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The Independent Set Problem Is FPT for Even-Hole-Free Graphs
International Symposium on Parameterized and Exact Computation, 2019The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is
Edin Husić +2 more
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Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable
Journal of Graph Theory, 2016Irena Penev
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