scholarly journals Matrices attaining the minimum semidefinite rank of a chordal graph

2013 ◽  
Vol 438 (10) ◽  
pp. 3804-3816 ◽  
Author(s):  
Naomi Shaked-Monderer
Keyword(s):  
Chordal Graph

Inverse M-matrices completions of then-chordal graph

2005 ◽  
Vol 82 (3) ◽  
pp. 275-288 ◽  
Author(s):  
Xijuan Guo ◽  
Huiping Yao ◽  
Fang Cheng
Keyword(s):  
Chordal Graph

scholarly journals Counting and Sampling Markov Equivalent Directed Acyclic Graphs

2019 ◽  
Vol 33 ◽  
pp. 7984-7991
Author(s):  
Topi Talvitie ◽  
Mikko Koivisto
Keyword(s):  
Equivalence Class ◽  
Computational Cost ◽  
Chordal Graph ◽  
Directed Acyclic Graphs ◽  
Desktop Computer ◽  
Bounded Degree ◽  
Acyclic Orientations ◽  
Uniform Sampling ◽  
Acyclic Graphs ◽  
Graph Size

Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2nn4) arithmetic operations, improving a previous superexponential upper bound. The second requires O(k!2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.

scholarly journals Finding a Maximum-Weight Convex Set in a Chordal Graph

2019 ◽  
Vol 23 (2) ◽  
pp. 167-190
Author(s):  
Jean Cardinal ◽  
Jean-Paul Doignon ◽  
Keno Merckx
Keyword(s):  
Convex Set ◽  
Chordal Graph ◽  
Maximum Weight

Powers of the Vertex Cover Ideal of a Chordal Graph

2011 ◽  
Vol 39 (10) ◽  
pp. 3753-3764 ◽  
Author(s):  
Fatemeh Mohammadi
Keyword(s):  
Vertex Cover ◽  
Chordal Graph

Coloring squares of graphs via vertex orderings

2020 ◽  
pp. 2050093
Author(s):  
Mehmet Akif Yetim
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Chordal Graph ◽  
Upper Bounds ◽  
Cocomparability Graph ◽  
Vertex Ordering

We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that [Formula: see text] is [Formula: see text]-colorable when [Formula: see text] is a cocomparability graph, [Formula: see text]-colorable when [Formula: see text] is a strongly orderable graph and [Formula: see text]-colorable when [Formula: see text] is a dually chordal graph, where [Formula: see text] is the maximum degree and [Formula: see text] = max[Formula: see text] is the multiplicity of the graph [Formula: see text]. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.

scholarly journals Characterizations and algorithmic applications of chordal graph embeddings

1997 ◽  
Vol 79 (1-3) ◽  
pp. 171-188 ◽  
Author(s):  
Andreas Parra ◽  
Petra Scheffler
Keyword(s):  
Chordal Graph ◽  
Graph Embeddings

Clique Partitions of Chordal Graphs

1993 ◽  
Vol 2 (4) ◽  
pp. 409-415 ◽  
Author(s):  
Paul Erdős ◽  
Edward T. Ordman ◽  
Yechezkel Zalcstein
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Split Graph ◽  
Threshold Graph

To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.

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