Given a graph G on a vertex set V , MaxCut is the problem of finding a partition of V so that the number of edges cut is as large as possible. MaxCut has a simple random approximation algorithm. Given a graph G = (V,E), output a random partition (R,B) of V . Specifically, for each vertex v ∈ V , put v in R or B with probability 1 2 . Theorem 3. The expected number of edges cut by this algorithm...

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph Γ, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of Γ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular...

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

Given a graph G = (V,E) and a positive integer t ≥ 2, the task in the vertex cover Pt (V CPt) problem is to find a minimum subset of vertices F ⊆ V such that every path of order t in G contains at least one vertex from F . The V CPt problem is NP-complete for any integer t ≥ 2. Recently, the authors presented a dynamic programming algorithm with runtime 4 · n that can solve the V CP3 problem in...

We describe a simple greedy algorithm whose input is a set P of vertices on a combinatorial surface S without boundary and that computes a shortest cut graph of S with vertex set P . (A cut graph is an embedded graph whose removal leaves a single topological disk.) If S has genus g and complexity n, the running-time is O(n log n+(g + |P |)n). This is an extension of an algorithm by Erickson and...

Let K` denote the complete graph on ` vertices. We prove that there is a constant c = c(`), such that whenever p ≥ n−c, with probability tending to 1 when n goes to infinity, every maximum K`-free subgraph of the binomial random graph Gn,p is (`− 1)partite. This answers a question of Babai, Simonovits and Spencer [BSS90]. The proof is based on a tool of independent interest: we show, for instan...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

Absfmcf-We consider the real-weight maximum cut of a planar graph. Given an undirected planar graph with real-valued weights associated with its edges, find a partition of the vertices into two nonemply sets such that the sum of the weights of the edges connecting the two sets is maximum. The conventional maximum cut and minimum cut problems assume nonnegative edge weights, and thus are special...

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