[Paper Review] On the Partition Dimension and the Twin Number of a Graph

Carmen Hernando, Mercè Ferrater Mora+1 more|arXiv (Cornell University)|Feb 29, 2016

Graph Labeling and Dimension Problems11 references3 citations

TL;DR

This paper re-evaluates the characterization of connected graphs of order $ n \geq 9 $ with partition dimension $ n-2 $, correcting a prior claim of 23 such graphs by showing there are exactly 15, due to isomorphisms and incorrect assumptions in earlier work. By integrating the twin number concept and analyzing twin classes, the authors refine the classification and prove that previously cited graphs actually have partition dimension $ n-3 $, not $ n-2 $, using structural graph theory and distance-based partition analysis.

ABSTRACT

A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u,v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G. In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n having partition dimension n-2. This set is formed by exactly 15 graphs, instead of 23, as was wrongly stated in the paper: "Discrepancies between metric dimension and partition dimension of a connected graph", published in Discrete Mathematics in 2008.

Motivation & Objective

• To correct the erroneous characterization of connected graphs with partition dimension $ n-2 $ for $ n \geq 9 $, which previously listed 23 graphs.
• To refine the classification of such graphs by incorporating the twin number, a structural invariant based on maximal sets of pairwise twin vertices.
• To prove that several graphs previously claimed to have partition dimension $ n-2 $ actually have partition dimension $ n-3 $, due to incorrect assumptions in prior work.
• To establish a precise characterization of all connected graphs of order $ n \geq 9 $ with partition dimension $ n-2 $, using twin class structure and distance-based partition analysis.
• To resolve discrepancies in the literature by identifying isomorphic graphs and correcting their inclusion in the classification.

Proposed method

• Introduces the twin number $ \tau(G) $ as the size of the largest twin class (maximal set of pairwise twin vertices), and uses it as a structural invariant to constrain possible graphs with high partition dimension.
• Applies the concept of locating partitions: a partition $ \Pi $ of $ V(G) $ is locating if each vertex has a unique distance vector to the parts of $ \Pi $, and the partition dimension $ \beta_p(G) $ is the minimum size of such a partition.
• Uses the bound $ \beta_p(G) \leq \beta(G) + 1 $, where $ \beta(G) $ is the metric dimension, to relate partition dimension to known invariants.
• Employs case analysis based on the twin number $ \tau(G) \in \{n-4, n-3, n-2\} $, analyzing the structure of $ W $-sets (twin sets) and their neighborhoods to classify graphs.
• Applies structural lemmas to eliminate graphs with $ W $-distinguishing vertices and to constrain the possible subgraphs induced on $ N(W)\setminus W $, using distance vectors to verify or disprove locating partitions.
• Uses explicit construction of partitions $ \Pi $ with $ k = n-3 $ parts to prove $ \beta_p(G) \leq n-3 $ for graphs like $ F_1 $, contradicting earlier claims of $ \beta_p(G) = n-2 $.

Experimental results

Research questions

• RQ1 How many connected graphs of order $ n \geq 9 $ actually have partition dimension $ n-2 $, and what is the correct characterization?
• RQ2 Why do several graphs previously claimed to have partition dimension $ n-2 $ in fact have partition dimension $ n-3 $?
• RQ3 How does the twin number $ \tau(G) $ constrain the structure of graphs with high partition dimension?
• RQ4 What structural properties must a graph satisfy to achieve partition dimension $ n-2 $, and how can these be systematically derived?
• RQ5 Can the classification of graphs with partition dimension $ n-2 $ be corrected by identifying isomorphic graphs and eliminating invalid constructions?

Key findings

• The set of connected graphs of order $ n \geq 9 $ with partition dimension $ n-2 $ consists of exactly 15 graphs, not 23 as previously claimed.
• Graphs such as $ F_1 \cong \overline{K_{n-3}} \vee (K_2 + K_1) $, $ F_3 \cong K_1 \vee (\overline{K_{n-3}} + K_2) $, and $ F_5 \cong \overline{K_{n-4}} \vee (P_3 + K_1) $ have partition dimension $ n-3 $, not $ n-2 $, contradicting earlier results.
• The graphs $ G_4 $ and $ G_6 $ in the prior classification are isomorphic, reducing the total number of distinct graphs from 23 to 22, and further analysis shows only 15 satisfy $ \beta_p(G) = n-2 $.
• For graphs with $ \tau(G) = n-2 $, the only possibilities are $ H_1 $ and $ H_2 $, both of which are confirmed to have partition dimension $ n-2 $.
• When $ \tau(G) = n-3 $, the structure of $ G[N(W)\setminus W] $ and the degrees of vertices in $ V\setminus N(W) $ determine the isomorphism class, leading to graphs $ H_3 $ through $ H_{10} $, all with $ \beta_p(G) = n-2 $.
• For $ \tau(G) = n-4 $, the only possible graphs with $ \beta_p(G) = n-2 $ are $ H_{11} $, $ H_{12} $, $ H_{13} $, $ H_{14} $, and $ H_{15} $, derived from path or cycle structures on $ N(W)\setminus W $, and confirmed via distance vector uniqueness.