A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path  $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...

This paper surveys the applications of Smarandache’s notion to graph theory appeared in International J.Math.Combin. from Vol.1,2008 to Vol.3,2009. In fact, many problems discussed in these papers are generalized in this paper. Topics covered in this paper include: (1)What is a Smarandache System? (2)Vertex-Edge Labeled Graphs with Applications: (i)Smarandachely k-constrained labeling of a grap...

This paper verifies a result of [9] concerning graphoidal structure of Shenoy’s notion of independence for Dempster-Shafer theory of belief functions. Shenoy proved that his notion of independence has graphoidal properties for positive normal valuations. The requirement of strict positive normal valuations as prerequisite for application of graphoidal properties excludes a wide class of DS beli...

A simple graphoidal cover of a semigraph is such that any two paths in have atmost one end vertex common. The minimum cardinality called the covering number and denoted by . acyclic an In this paper we find for wheel semigraph, unicycle zero-divisor graph.

In any graph G = (V,E) that is not necessarily finite, a graphoidal cover is a set ψ of nontrivial paths P1, P2, . . . , not necessarily open and called ψ -edges, such that (GC-1) no vertex of G is an internal vertex of more than one path in ψ, and (GC-2) every edge of G is in exactly one of the paths in ψ. A ψ -dominating set of G is then defined as a set D of vertices in G such that every ver...

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...