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Clique irreducibility of some iterative classes of graphs

Aparna Lakshmanan S.A. Vijayakumar — 2008

Discussiones Mathematicae Graph Theory

In this paper, two notions, the clique irreducibility and clique vertex irreducibility are discussed. A graph G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and it is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G. It is proved that L(G) is clique irreducible if and only if every triangle in G has a vertex of degree two. The conditions for the iterations of line graph,...

The edge C₄ graph of some graph classes

Manju K. MenonA. Vijayakumar — 2010

Discussiones Mathematicae Graph Theory

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...

Power Domination in Knödel Graphs and Hanoi Graphs

Seethu VargheseA. VijayakumarAndreas M. Hinz — 2018

Discussiones Mathematicae Graph Theory

In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs [...] Hpn H p n . We determine the power domination number of W3,2ν and provide an upper bound for the power domination number of Wr+1,2r+1 for r ≥ 3. We also compute the k-power domination number and the k-propagation radius of [...] Hp2 H p 2 .

Containers and wide diameters of P 3 ( G )

Daniela FerreroManju K. MenonA. Vijayakumar — 2012

Mathematica Bohemica

The P 3 intersection graph of a graph G has for vertices all the induced paths of order 3 in G . Two vertices in P 3 ( G ) are adjacent if the corresponding paths in G are not disjoint. A w -container between two different vertices u and v in a graph G is a set of w internally vertex disjoint paths between u and v . The length of a container is the length of the longest path in it. The w -wide diameter of G is the minimum number l such that there is a w -container of length at most l between any pair of different...

Gallai and anti-Gallai graphs of a graph

S. Aparna LakshmananS. B. RaoA. Vijayakumar — 2007

Mathematica Bohemica

The paper deals with graph operators—the Gallai graphs and the anti-Gallai graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs and the anti-Gallai graphs to be H -free for any finite graph H . The case of complement reducible graphs—cographs is discussed in detail. Some relations between the chromatic number, the radius and the diameter of a graph and its Gallai and anti-Gallai graphs are also obtained.

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