On the stress transit function

TL;DR

This work introduces the stress transit framework on graphs, defining the stress interval $S_G(u,v)$ as the set of vertices that lie on every shortest $u$-$v$ path and developing the corresponding notions of $s$-convexity, $s$-extreme vertices, the stress number $sn(G)$, and the stress hull number $sh(G)$. It provides structural characterizations, including that $S(u,v)$ is always $s$-convex and can be described via cut-vertices of $G[I(u,v)]$, and establishes that $G$ is geodetic iff $S(u,v)=I(u,v)$. The authors analyze and compare the invariants across graph families, proving that split graphs satisfy $sh(G)=sn(G)=|Ext_s(G)|$, while constructing graphs with arbitrarily large gaps between these measures; they also show that computing stress intervals is polynomial-time and that the stress-set decision problem is NP-complete, even on bipartite graphs. The work closes with discussions of implications for graph products and block graphs, and outlines several open questions, including the complexity of stress-set related problems on chordal graphs and questions about the stress convexity number. Overall, the paper lays foundational properties, computational aspects, and exact values for key invariants in the novel stress convexity framework, with potential implications for network analysis and related graph convexities.

Abstract

The stress interval $S(u,v)$ between $u,v\in V(G)$ is the set of all vertices in a graph $G$ that lie on every shortest $u,v$-path. A set $U \subseteq V(G)$ is stress convex if $S(u,v) \subseteq U$ for any $u,v\in U$. A vertex $v \in V(G)$ is s-extreme if $V(G)-v$ is a stress convex set in $G$. The stress number $sn(G)$ of $G$ is the minimum cardinality of a set $U$ where $\bigcup_{u,v \in U}S(u,v)=V(G)$. 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 $G$, $sh(G)=sn(G)=|Ext_s(G)|$, where $Ext_s(G)$ is the set of s-extreme vertices of $G$. Finally, we show that for $k \in \mathbb{N}$, deciding whether $sn(G) \leq k$ is NP-complete problem, even when restricted to bipartite graphs.

Figures (3)

• Figure 1: A graph $G_{4,5}$.
• Figure 2: A graph $C_5\Box C_5$ with a stress set (black vertices).
• Figure 3: Construction of a bipartite graph $G'$ from a bipartite graph $G$.

Theorems & Definitions (49)

• Remark 1
• Proposition 2
• proof
• Theorem 3
• Theorem 4
• proof
• Lemma 5
• proof
• Proposition 6
• proof