Edge total mutual-visibility in graphs

Let G be a graph with vertex set V(G) and edge-set E(G). The total mutual-visibility number of G, denoted by μt(G), is the order of a largest set A⊆V(G) such that for each two vertices x and y of G, there exists a shortest x, y-path Px,y with V(Px,y)∩A⊆{x,y}. In this paper, we introduce the edge analogue of the total mutual-visibility number of G. The edge total mutual-visibility number of G, denoted by eμt(G), is the order of a largest set A⊆E(G) such that for each two edges e and e′ of G, there exists a shortest e,e′-path Pe,e′ with E(Pe,e′)∩A=∅. We determine the edge total mutual-visibility number for several families of graphs that include for instance the cycle Cn, the complete bipartite graph Km,n, and the wheel graph W1,n. In addition, we study this invariant under the join of graphs, hierarchical product, corona and edge corona product of two graphs.