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 Gallai graphs, the anti-Gallai graphs and its iterations to be clique irreducible and clique vertex irreducible are also obtained.
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.
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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 $H_p^2$ .
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