Odd twists on strongly chordal graphs

2019 ◽  
Vol 11 (03) ◽  
pp. 1950034
Author(s):  
Terry A. McKee
Keyword(s):  
Chordal Graphs ◽  
Odd Cycle ◽  
Strongly Chordal Graphs

Strongly chordal graphs can be characterized as chordal graphs in which every even cycle of length at least [Formula: see text] has an odd chord (a chord whose endpoints are an odd distance apart in the cycle subgraph). Define “oddly chordal graphs” to be chordal graphs in which every odd cycle of length at least [Formula: see text] has an odd chord. Strongly chordal graphs are shown to be oddly chordal, and the oddly chordal graphs are characterized by forbidding induced “double [Formula: see text]-sun” subgraphs. Both strongly chordal and oddly chordal graphs are also characterized in terms of uncrossed chords of appropriate-length cycles.

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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