Some characterization theorems for the discrete holometric space.
On order and metric convexities in
Equienergetic self-complementary graphs
In this paper equienergetic self-complementary graphs on vertices for every , and , are constructed.
Clique irreducibility of some iterative classes of graphs
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
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...
Some properties of the discrete holometric space
Power Domination in Knödel Graphs and Hanoi Graphs
In this paper, we study the power domination problem in Knödel graphs WΔ,2ν and Hanoi graphs [...] Hpn . 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 .
Containers and wide diameters of
The intersection graph of a graph has for vertices all the induced paths of order 3 in . Two vertices in are adjacent if the corresponding paths in are not disjoint. A -container between two different vertices and in a graph is a set of internally vertex disjoint paths between and . The length of a container is the length of the longest path in it. The -wide diameter of is the minimum number such that there is a -container of length at most between any pair of different...
Gallai and anti-Gallai graphs of a graph
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 -free for any finite graph . 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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