### Prabhakar and Hilfer-Prabhakar Fractional Derivatives in the Setting of $\Psi$-Fractional Calculus and Its Applications

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**Authors:**S. K. MAGAR, P. V. DOLE AND K. P. GHADLE

**DOI:**10.46793/KgJMat2105.685K

**Abstract:**

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γ

_{veR}(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We ﬁrst show that determining the number γ

_{veR}(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree diﬀerent from a star with order n, l leaves and s support vertices, then γ

_{veR}(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γ

_{veR}(T) = 2γ

^{′}(T), where γ

^{′}(T) is the edge domination number of T.

**Keywords:**

Vertex-edge roman dominating set, edge dominating set, trees.

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