Metric Location in Pseudotrees: A survey and new results

Abstract

The aim of this paper is to revise the literature on different metric locations in the families of paths, cycles, trees and unicyclic graphs, as well as, provide several new results on that matter.

Figures (12)

• Figure 1: (White) square vertices: (Strong) support vertices. The white circle is an exterior major vertex but it is neither strong nor a support vertex. Triangle vertices are both major vertices but none of them is neither an exterior major vertex nor a support vertex. Check that $g=8$, $\ell=10$, $\lambda=6$, $\ell_s=8$, $\lambda_s=4$, $c_2=3$, $c_3=5$, $\rho=3$.
• Figure 2: Left: Odd unicyclic graph $G$ with $\ell(G)=4$, $\lambda(G)=2$, $\rho(G) =2$, without antipodal branch-active vertices. Right: Odd unicyclic graph $G'$ with $\ell(G)=5$, $\lambda(G)=3$, $\rho(G) =2$, without antipodal branch-active vertices.
• Figure 3: Several situations of the proof of Proposition \ref{['lem:evencycle.rho=0']}
• Figure 4: Even unicyclic graph $G$ with $g-2$ threads and $\dim(G)=2$.
• Figure 5: Left: Even unicyclic graph $G$ with $\ell(G)=2$, $\lambda(G)=1$, $\rho(G) =1$. Right: Even unicyclic graph $G'$ with $\ell(G')=3$, $\lambda(G')=2$, $\rho(G') =1$.

Theorems & Definitions (80)

• Definition 1
• Theorem 2
• Theorem 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Definition 6
• Lemma 7