Path-Induced Closed Geodetic Numbers of Some Graphs

Path-Induced Closed Geodetic Domination of Some Common Graphs and Edge Corona of Graphs

“…[3] Every cut-vertex of a connected graph G belongs to every path-induced closed geodetic set of G. Theorem 3. [3] Let G be a connected graph with cut-vertices. If G admits path-induced closed geodetic set, then G ∖ x has exactly two components for each cut-vertex x of G. Theorem 4.…”

“…If G admits path-induced closed geodetic set, then G ∖ x has exactly two components for each cut-vertex x of G. Theorem 4. [3] Let G be a connected graph with cut-vertices. If G admits a path-induced closed geodetic set, then each block of G admits at most 2 cut-vertices.…”

“…Theorem 5. [3] Let G be a connected graph of order m such that G admits a path-induced closed geodetic set. If every vertex of G is either an extreme vertex or a cut-vertex, then picgn (G) = m. Theorem 6.…”

“…On the other hand, O. Cauntongan and I. Aniversario [3] introduced and studied the concept on path-induced closed geodetic number of some graphs. This concept follows from the definition of geodetic numbers of graphs introduced by Buckley and Harary in [2], closed geodetic numbers in [1], and path-induced geodetic numbers in [7].…”

“…Definition 9. [3] Let G be a connected graph of order n and S ⊆ V (G). A closed geodetic cover S of G is called a path-induced closed geodetic set of a graph G, denoted by picgset, if ⟨S⟩ has a Hamiltonian path.…”