In 1994 S. McGuinness showed that any greedy clique decomposition of an n-vertex graph has at most ⌊n2/4⌋ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empt...

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 domina...

In this paper, we introduce a new graph theoretic parameter, split edge geodetic domination number of connected as follows. A set S ⊆ V(G) is said to be dominating G if both and ( < V-S > disconnected). The minimum cardinality the called denoted by γ1gs(G). It shown that for any 3 positive integers m, f nwith 2 ≤ m n-2, there exists order n such g1 (G) = γ1gs f. For every pair l, with l γ1gs(G)...

Foucaud, Krishna and Ramasubramony Sulochana recently introduced the concept of monitoring edge-geodetic sets in graphs, a related graph invariant. These are vertices such that removal any edge changes distance between some pair set. They studied minimum possible size set given graph, which we call number. We show decision problem for number is NP-complete. also give best-possible upper lower b...

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi) graph is a simple, complete, edge-colored (multi) graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decompos...

By using seismic records of the 2004 magnitude 6.0 Parkfield earthquake, we identified a burst of high-frequency seismic radiation that occurred about 13 kilometers northwest of the hypocenter and 5 seconds after rupture initiation. We imaged this event in three dimensions by using a waveform back-projection method, as well as by timing distinct arrivals visible on many of the seismograms. The ...

Proof: “⇒” Suppose D = (V,A) is a strongly connected graph obtained as an orientation of G = (V,E). Then, since ∀S ⊂ V , |δ D(S)| ≥ 1 and |δ − D(S)| ≥ 1, we have |δG(S)| ≥ 2. Therefore, G is 2-edge-connected. “⇐” Let G be a 2-edge-connected graph. Then G has an ear-decomposition. In other words, G is either a cycle C or G is obtained from a 2-edge-connected graph G by adding an ear P (a path) c...