Strengthening strongly chordal graphs

2016 ◽  
Vol 08 (01) ◽  
pp. 1650002
Author(s):  
Terry A. McKee
Keyword(s):  
Chordal Graphs ◽  
Strongly Chordal Graphs

An [Formula: see text]-chord of a cycle [Formula: see text] is a chord that forms a new cycle with a length-[Formula: see text] subpath of [Formula: see text] when [Formula: see text] is at most half the length of [Formula: see text]. Define a graph to be [Formula: see text]-strongly chordal if, for every [Formula: see text], every cycle long enough to have an [Formula: see text]-chord always has an [Formula: see text]-chord. The [Formula: see text]-strongly chordal and [Formula: see text]-strongly chordal graphs are, respectively, the chordal and strongly chordal graphs. Several characterizations of [Formula: see text]-strongly chordal graphs are proved, along with details of the class of [Formula: see text]-strongly chordal graphs.

Partitioning Cliques of Claw-Free Strongly Chordal Graphs

1999 ◽  
pp. 142-151 ◽  
Author(s):  
G. Confessore ◽  
P. Dell’Olmo ◽  
S. Giordani
Keyword(s):  
Chordal Graphs ◽  
Strongly Chordal Graphs

Symmetric graph-theoretic roles of two-pairs and chords of cycles

2014 ◽  
Vol 06 (03) ◽  
pp. 1450031
Author(s):  
Terry A. McKee
Keyword(s):  
Chordal Graphs ◽  
Symmetric Graph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
Strongly Chordal Graphs

Although the notion of a two-pair (a pair of vertices between which all induced paths have length 2) was invented for the class of weakly chordal graphs, two-pairs can also play a fundamental role for smaller graph classes. Indeed, two-pairs and chords of cycles can collaborate symmetrically to give parallel characterizations of weakly chordal, chordal, and strongly chordal graphs (and of distance-hereditary graphs).

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals Cycle Extendability of Hamiltonian Strongly Chordal Graphs

2021 ◽  
Vol 35 (3) ◽  
pp. 2115-2128
Author(s):  
Guozhen Rong ◽  
Wenjun Li ◽  
Jianxin Wang ◽  
Yongjie Yang
Keyword(s):  
Chordal Graphs ◽  
Strongly Chordal Graphs ◽  
Cycle Extendability
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