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10
On Minimal Prime Extensions of a FourVertex Graph in a Prime Graph
, 2002
"... In a finite undirected graph G = (V; E), a homogeneous set is a set U V of at least two vertices such that every vertex in V n U is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimum... ..."
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In a finite undirected graph G = (V; E), a homogeneous set is a set U V of at least two vertices such that every vertex in V n U is either adjacent to all vertices of U or nonadjacent to all of them. A graph is prime if it does not have a homogeneous set. We investigate the minimum...
Efficient Robust Algorithms for the Maximum Weight Stable Set Problem in Chairfree Graph Classes
, 2001
"... Modular decomposition of graphs is a powerful tool for designing efficient algorithms for algorithmic graph problems such as the Maximum Weight Stable Set Problem and the Maximum Weight Clique Problem. Using this tool we obtain O(nm) time algorithms for the Maximum Weight Stable Set Problem on (chai ..."
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Cited by 3 (1 self)
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Modular decomposition of graphs is a powerful tool for designing efficient algorithms for algorithmic graph problems such as the Maximum Weight Stable Set Problem and the Maximum Weight Clique Problem. Using this tool we obtain O(nm) time algorithms for the Maximum Weight Stable Set Problem on (chair, coP)free, (chair,P5)free and (chair,bull)free graphs. Moreover, our algorithms are robust in the sense that we do not have to check in advance whether the input graphs are indeed (chair, coP)free or (chair,P5)free or (chair,bull)free.
The Reducing Copath Method for Simple Homogeneous Sets
, 2001
"... A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension ..."
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Cited by 2 (1 self)
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A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)nfHg. We denote by Ext(G) the set of all minimal extensions of a graph G. Zverovich [4]
A Finiteness Theorem for Primal Extensions
, 2001
"... A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension H ..."
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Cited by 1 (0 self)
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A set W ` V (G) is called homogeneous in a graph G if 2 jW j jV (G)j \Gamma 1, and N(x)nW = N(y)nW for each x; y 2 W . A graph without homogeneous sets is called prime. A graph H is called a (primal) extension of a graph G if G is an induced subgraph of H , and H is a prime graph. An extension H of G is minimal if there are no extensions of G in the set ISub(H)nfHg. We denote by Ext(G) the set of all minimal extensions of a graph G.
The Reducing Copath Method for Specific Homogeneous Sets
, 2001
"... We specify Reducing Copath Method of Zverovich [4] for some types of homogeneous sets. ..."
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Cited by 1 (1 self)
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We specify Reducing Copath Method of Zverovich [4] for some types of homogeneous sets.
A Characterization of Domination Reducible Graphs
, 2001
"... We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs. RRR 232001 Page 1 1 ..."
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We introduce a hereditary class of domination reducible graphs where the minimum dominating set problem is polynomially solvable, and characterize this class in terms of forbidden induced subgraphs. RRR 232001 Page 1 1
Extended ...Free Graphs
, 2001
"... Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edgeset fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as th ..."
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Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edgeset fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as the class P consisting of all graphs which can be obtained from graphs in P by repeated substitutions. We characterize the substitutional closure (P 5 [ K 1 ; P 5 [ K 1 )free graphs in terms of forbidden induced subgraphs. RRR 222001 Page 1 1
Extension of ClawFree Graphs and ...Free Graphs With Substitutions
, 2001
"... Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edgeset fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as th ..."
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Let G and H be graphs. A substitution of H in G instead of a vertex v 2 V (G) is the graph G(v ! H), which consists of disjoint union of H and G \Gamma v with the additional edgeset fxy : x 2 V (H); y 2 NG (v)g. For a hereditary class of graphs P , the substitutional closure of P is defined as the class P consisting of all graphs which can be obtained from graphs in P by repeated substitutions. We give forbidden induced subgraph characterizations of the substitutional closure of Clawfree graphs, and (K 1 [ P 4 )free graphs. Note that the weighted stability number problem can be solved in polynomial time for (Clawfree graphs) , and ((K 1 [ P 4 )free graphs) . Also, these classes are polynomially recognizible.
Substitutionclosed pattern classes
"... The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set o ..."
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The substitution closure of a pattern class is the class of all permutations obtained by repeated substitution. The principal pattern classes (those defined by a single restriction) whose substitution closure can be defined by a finite number of restrictions are classified by listing them as a set of explicit families.
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 43 (2009), Pages 163–180 On split and almost CISgraphs ∗
"... A CISgraph is defined as a graph whose every maximal clique and stable set intersect. These graphs have many interesting properties, yet, it seems difficult to obtain an efficient characterization and/or polynomialtime recognition algorithm for CISgraphs. An almost CISgraph is defined as a graph ..."
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A CISgraph is defined as a graph whose every maximal clique and stable set intersect. These graphs have many interesting properties, yet, it seems difficult to obtain an efficient characterization and/or polynomialtime recognition algorithm for CISgraphs. An almost CISgraph is defined as a graph that has a unique pair (C, S) of disjoint maximal clique C and stable sets S. We conjecture that almost CISgraphs are exactly split graphs that have a unique split partition and prove this conjecture for a large hereditary class of graphs that contains, for example, chordal graphs and P5free graphs, as well as their complements, etc. We also prove the conjecture in case C  = S  =2 and show that the vertexset R = V \ (C ∪ S) cannot induce a threshold graph, although we do not prove that R = ∅, as the conjecture suggests. 1