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A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs

100%
Metelsky Y., Schemeleva K., Werner F.
Discussiones Mathematicae Graph Theory
|
2017
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tom 37
|
nr 1
13-28
EN
We characterize the class [...] L32 $L_3^2 $ of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from [...] L32 $L_3^2 $ in the class of threshold graphs, where n is the number of vertices of a tested graph.
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The edge C₄ graph of some graph classes

100%
Menon M., Vijayakumar A.
Discussiones Mathematicae Graph Theory
|
2010
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tom 30
|
nr 3
365-375
EN
The edge C₄ graph of a graph G, E₄(G) is a graph whose vertices are the edges of G and two vertices in E₄(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C₄. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C₄ graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E₄(G). It is shown that for any graph G without isolated vertices, there exists a super graph H such that C(H) = G and C(E₄(H)) = E₄(G). Also we give forbidden subgraph characterizations for E₄(G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.
3
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The i-chords of cycles and paths

86%
McKee T.
Discussiones Mathematicae Graph Theory
|
2012
|
tom 32
|
nr 4
607-615
EN
An i-chord of a cycle or path is an edge whose endpoints are a distance i ≥ 2 apart along the cycle or path. Motivated by many standard graph classes being describable by the existence of chords, we investigate what happens when i-chords are required for specific values of i. Results include the following: A graph is strongly chordal if and only if, for i ∈ {4,6}, every cycle C with |V(C)| ≥ i has an (i/2)-chord. A graph is a threshold graph if and only if, for i ∈ {4,5}, every path P with |V(P)| ≥ i has an (i -2)-chord.
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