Convexity in graphs

Computing the Shortest Essential Cycle

“…A subset X of a (not necessarily combinatorial) surface Σ is convex if every shortest path in Σ between two points in X lies entirely in X . This definition is consistent with both Harary and Nieminen's definition of geodesic convexity in graphs [3,22] and the standard definition of total convexity in Riemannian geometry [1]. In particular, a cycle is convex if and only if no shortest path crosses the cycle more than once; moreover, a convex cycle must be simple (not just essentially simple).…”

The Forcing Convexity Number of a Graph

“…Convexity in graphs was also studied in [2,3,4]. For a connected graph G of order at least 3, the convexity number con(G) of G was defined in [2] as the maximum cardinality of a proper convex set of G, that is, con(G) = max {|S| : S is a convex set of G and S = V (G)} .…”

$1$-Movable Clique Dominating Sets of a Graph

“…1-movable convex domination number ) of G. A 1-movable clique dominating set in G with cardinality γ 1 mcl (G) is called a γ 1 mcl -set of G. The concept of clique domination is studied by Cozzens and Kelleher in [4]. Convexity and some of its related concepts are studied and investigated in [2], [3], and [5] while convex domination in graphs are dealt with in [8] and [9]. Blair et al introduced and studied movable domination in [1].…”