Completeness-resolvable graphs

TL;DR

This paper introduces the concept of completeness-resolvable graphs, characterizes their structure using bipartite graph edge coverings, and establishes bounds for minimal graphs with such properties.

Contribution

It constructs the set of all completeness-resolvable graphs, organizes them into posets, and characterizes minimal graphs that meet edge bounds.

Findings

Constructed the set of all completeness-resolvable graphs.

Established posets based on spanning subgraph relationships.

Characterized minimal graphs satisfying edge bounds.

Abstract

Given a connected graph $G=(V(G), E(G))$, the length of a shortest path from a vertex $u$ to a vertex $v$ is denoted by $d(u,v)$. For a proper subset $W$ of $V(G)$, let $m(W)$ be the maximum value of $d(u,v)$ as $u$ ranging over $W$ and $v$ ranging over $V(G)\setminus W$. The proper subset $W=\{w_1,\ldots,w_{|W|}\}$ is a {\em completeness-resolving set} of $G$ if $$ \Psi_W: V(G)\setminus W \longrightarrow [m(W)]^{|W|},\qquad u\longmapsto (d(w_1,u),\ldots,d(w_{|W|},u)) $$ is a bijection, where $$ [m(W)]^{|W|}=\{(a_{(1)},\ldots,a_{(|W|)})\mid 1\leq a_{(i)}\leq m(W)\text{ for each }i=1,\ldots,|W|\}. $$ A graph is {\em completeness-resolvable} if it admits a completeness-resolving set. In this paper, we first construct the set of all completeness-resolvable graphs by using the edge coverings of some vertices in given bipartite graphs, and then establish posets on some subsets of this set by the spanning subgraph relationship. Based on each poset, we find the maximum graph and give the lower and upper bounds for the number of edges in a minimal graph. Furthermore, minimal graphs satisfying the lower or upper bound are characterized.