By applying the Tutte decomposition of 2{connected graphs into 3{block trees we provide unique structural characterizations of several classes of 2{connected graphs, including minimally 2{connected graphs, minimally 2{edge{connected graphs, critically 2{connected graphs, critically 2{edge{connected graphs, 3{edge{connected graphs, 2{connected cubic graphs and 3{connected cubic graphs. We also g...

The vertex-edge Wiener index of a simple connected graph G is defined as the sum of distances between vertices and edges of G. Two possible distances D_1(u,e|G) and D_2(u,e|G) between a vertex u and an edge e of G were considered in the literature and according to them, the corresponding vertex-edge Wiener indices W_{ve_1}(G) and W_{ve_2}(G) were introduced. In this paper, we present exact form...

A (p, q) graph G is called edge-magic if there exists a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , p + q} such that f(u) + f(v) + f(uv) is constant for any edge uv of G. Moreover, G is said to be super edgemagic if f(V (G)) = {1, 2, . . . , p}. Every super edge-magic (p, q) graph is harmonious, sequential and felicitous whenever it is a tree or satisfies q ≥ p. In this paper, we prove...

Numerous networks as, for example, road networks, electrical networks and communication networks can be modeled by a graph. Many attempts have been made to determine how well such a network is "connected" or stated differently how much effort is required to break down communication in the system between at least some nodes. Two well-known measures that indicate how "reliable" a graph is are the...

A (p,q) graph G is called super edge-magic if there exists a bijective function f from V (G) ∪ E(G) to {1, 2,. .. , p + q} such that f (x) + f (xy) + f (y) is a constant k for every edge xy of G and f (V (G)) = {1, 2,. .. , p}. Furthermore, the super edge-magic deficiency of a graph G is either the minimum nonnegative integer n such that G ∪ nK 1 is super edge-magic or +∞ if there exists no suc...

For a connected graph G = (V, E), an edge set S ⊆ E is a k-restricted edge cut if G−S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G) = min{|[X, X ]| : |X | = k,G[X ] is connected}. G is λk -optimal if λk(G) = ξk(G). Moreover, G is super-λk ...

We show that for every ℓ , there exists d such 3-edge-connected graph with minimum degree can be edge-partitioned into paths of length (provided its number edges is divisible by ). This improves a result asserting 24-edge-connectivity and high provides partition. best possible as 3-edge-connectivity cannot replaced 2-edge connectivity.