In the 4-Path Vertex Cover problem, input is an undirected graph G and integer k . The goal to decide whether there a set S of vertices size at most such that every path with 4 in contains least one vertex this paper we give parameterized algorithm for whose time complexity 2 61 9 ? n O ( 1 ) , where denotes number graph.

For a given undirected graph G, the maximum multiplicity of G is defined to be the largest multiplicity of an eigenvalue over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i 6= j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertex-disjoint paths occurring as induced subgraphs of G that cover all the vertices of G. We derive a formula f...

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ-hyperbolic if every geodesic triangle is δ-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a cha...

Before surveying the results of the paper, we introduce path algebras. Path algebras play a central role in the representation theory of finite-dimensional algebras (Gabriel, 1980; Auslander et al., 1995; Bardzell, 1997) and the theory of Gröbner bases (Bergman, 1978; Mora, 1986; Farkas et al., 1993) has been an important tool in some results (Feustel et al., 1993; Green and Huang, 1995; Bardze...

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

In the 4-Path Vertex Cover problem, input is an undirected graph G and integer k . The goal to decide whether there a set S of vertices size at most such that every path with 4 in contains least one vertex this paper we give parameterized algorithm for whose time complexity 2 61 9 ⋅ n O ( 1 ) , where denotes number graph.

We consider economical representations for the path information in a directed graph. A directed graph G is said to be a transitive reduction of the directed graph G provided that (i) G has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than G satisfying condition (i). Though directed graphs wit...

A graph G = (V,E), where V is the set of vertices, and E is the set of edges. An edge e ∈ E is an unordered pair (u,v) in undirected graphs, where u,v∈V . In directed graphs, an edge e is an ordered pair. A path from a vertex u to a vertex v is a sequence of vertices (w0,w1, ...wk), where u = w0, v = wk and (wi−1,wi) ∈ E for all 1 ≤ i ≤ k. The path is a cycle if u = v. The length of a path in a...

The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O(kn) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k ...

We are given an acyclic directed graph with one source, and a subset of its edges which contains exactly one outgoing edge for every non-sink vertex. These edges determine a unique path from the source to a sink. We can think of it as a switch in every vertex, which determines which way the water arriving to that vertex flows further. We are interested in determining either the sink the flow ar...