The strong vertex span of trees

TL;DR

The paper studies the strong vertex span of trees, defined as $σ^{\boxtimes}_V(G)$, representing the maximum distance two independent walkers can maintain while visiting all vertices via surjective weak homomorphisms from a path. It introduces the triod size $\eta(v)$ and the tree-wide measure $\mathfrak{H}(T)$, proves the key bound $σ^{\boxtimes}_V(T) \le \mathfrak{H}(T)$ for non-path trees, and develops rooted-tree lemmas that justify evaluating spans at a central vertex. It then presents a linear-time algorithm that roots the tree at a center $c$, computes $\mathfrak{R}(C_2(v))$ and $\mathfrak{R}(C_3(v))$ via a DFS-based height computation, and uses $η(v)=\mathfrak{H}(T)$ to determine $σ^{\boxtimes}_V(T)$. Consequently, the strong vertex and strong edge spans coincide for trees, and the approach yields a practical, linear-time solution for trees, contrasting with the general $O(n^4)$-time bounds for arbitrary graphs.

Abstract

The strong vertex (edge) span of a given graph $G$ is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of $G$ and moving either to an adjacent vertex or staying in the current position independently of each other. We introduce the notions of switching walks and triod size of a tree, which are used to determine the strong vertex and the strong edge span of an arbitrary tree. The obtained results are used in an algorithm that computes the strong vertex (edge) span of the input tree in linear time.

Figures (7)

• Figure 1: Case 1 when $A(t')\in C_{\beta}(\widehat{v})$.
• Figure 2: Case 2 when $\widetilde{v}\in C_{\alpha}(\widehat{v})$.
• Figure 3: A sketch of the graph structure for the Case 3.
• Figure 4: Situation when $\widetilde{v}\in C_{\beta}(\widehat{v})$ and $A(\widetilde{t}-1)$ is in the path from $\widehat{B}$ to $\widehat{v}$ (Subcase 4.1.).
• Figure 5: Situation when $C_{\widehat{\alpha}}(\widetilde{v})\not= C_{\widehat{\delta}}(\widetilde{v})$ and $C_{\widehat{\alpha}}(\widetilde{v})= C_{\widehat{\beta}}(\widetilde{v})$, one possibility of Subcase 4.2.

Theorems & Definitions (27)

• Definition 1.1: BaTa23
• Remark 1.2
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Definition 2.5
• Theorem 2.6