The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph G and seeks smallest vertex set S that hits all cycles in . This is one of Karp’s 21 NP -complete problems. Resolving the parameterized complexity status DFVS was long-standing open until Chen et al. (2008) showed its fixed-parameter tractability via 4 k ! n O ( 1 ) -time algorithm, where = | Here we show two genera...

A topological index is a function having set of graphs as its domain and real numbers range. Here we concentrated on indices involving the number vertices, edges maximum minimum vertex degree. The aim this paper to compute lower upper bounds second Zagreb index, third Hyper Harmonic Redefined first First reformulated Forgotten square F-index, Sum-connectivity Randic Reciprocal Gourava Sombar Ni...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....