Given a network G = (V,E), we say that a subset of vertices S ⊆ V has radius r if it is spanned by a tree of depth r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since in the special case r = 0, a set spanned by a tree of depth r is a single vertex. Our motivation for consid...

Let G be a connected graph with minimum degree δ and edgeconnectivity λ. A graph is maximally edge-connected if λ = δ, and it is superedge-connected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is maximally edge-connected or sup...

In this we prove that every 3-connected planar graph has closed walk each vertex, none more than twice, such that any vertex visited twice is in a vertex cut of size 3. This both Tutte's Theorem that 4-connected planar graphs are Hamiltonian and the result of Gao and Richter that 3-connected planar graphs have a closed walk visiting each vertex at least once but at most twice.

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...

A 4-restricted edge cut is an edge cut of a connected graph which disconnects the graph, where each component has order at least 4. Graphs that contain 4-restricted edge cuts are characterized in this paper. As a result, it is proved that a connected graph G of order at least 10 contains 4-restricted edge cuts if and only if it contains no cut-vertex u where every component of G \ u has order a...

The variable sum exdeg index of a graph G is defined as $SEI_a(G)=sum_{uin V(G)}d_G(u)a^{d_G(u)}$, where $aneq 1$ is a positive real number,  du(u) is the degree of a vertex u ∈ V (G). In this paper, we characterize the graphs with the extremum variable sum exdeg index among all the graphs having a fixed number of vertices and cut edges, for every a>1.

For a connected graph G = (V (G), E(G)), a vertex set S ⊆ V (G) is a k-restricted vertex-cut if G − S is disconnected such that every component of G − S has at least k vertices. The k-restricted connectivity κk(G) of the graph G is the cardinality of a minimum k-restricted vertex-cut of G. In this paper, we give the 3-restricted connectivity and the 4-restricted connectivity of the Cartesian pr...

We propose Inflate-Paste – a new technique for generating orthogonal polygons with a given number of vertices from a unit square based on gluing rectangles. It is dual to Inflate-Cut – a technique we introduced in [12] that works by cutting rectangles.

Given a graph G, we give a fast algorithm for approximating the vertex connectivity κ of G. Our algorithm delivers a minimum vertex cut of G if κ <= δ/2 , and returns a message “κ > δ/2 ” otherwise, where δ denotes the minimum degree of G. The algorithm runs in O(n2(1+min{κ2, κ√n}/δ)) time and O(n+m) space, where n and m denote the numbers of vertices and edges in G, respectively. key words: gr...

Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if κ(G− v) > κ(G), where κ(G) is the number of components in G. Similarly, an edge e is a bridge of G if κ(G− e) > κ(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can b...