On the Stress Transit Function

Abstract The stress interval S (u, v) between u, v V (G) u, v ∈ V (G) is the set of all vertices in a graph G that lie on every shortest u, v -path. A set U V (G) U ⊆ V (G) is stress convex if S (u, v) U S (u, v) ⊆ U for any u, v U u, v ∈ U. A vertex v V (G) v ∈ V (G) is s-extreme if V (G) -\v\ V (G) - v is a stress convex set in G. The stress number sn (G) of G is the minimum cardinality of a set U where ₔ, ₕ ₔS (u, v) =V (G) ⋃ u, v ∈ U S (u, v) = V (G). The stress hull number sh (G) of G is the minimum cardinality of a set whose stress convex hull is V (G). In this paper, we present many basic properties of stress intervals. We characterize s-extreme vertices of a graph G and construct graphs G with arbitrarily large difference between the number of s-extreme vertices, sh (G) and sn (G). Then we study these three invariants for some special graph families, such as graph products, split graphs, and block graphs. We show that in any split graph G, sh (G) =sn (G) =|Extₛ (G) | s h (G) = s n (G) = | Ext