https://orbilu.uni.lu/bitstream/10993/40570/1/omsdim-published-version.pdf

Distance-based vertex identification in graphs: The outer multiset dimension

<jats:p>For a set <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> of vertices and a vertex <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"><mml:mi>v</mml:mi></mml:math> of a connected graph <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\"><mml:mi>G</mml:mi></mml:math>, the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\"><mml:mi>k</mml:mi></mml:math>-multiset <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\"><mml:mi>m</mml:mi><mml:mi>r</mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v</mml:mi><mml:mfenced open=\"|\" close=\"\" separators=\"|\"><mml:mrow><mml:mi>W</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced open=\"{\" close=\"}\" separators=\"|\"><mml:mrow><mml:mi>d</mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\"><mml:mi>d</mml:mi><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> is the distance from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\"><mml:mi>v</mml:mi></mml:math> to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\"><mml:msub><mml:mrow><mml:mi>w</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:math> for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\"><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1,2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:math>, and is the <jats:italic>multirepresentation</jats:italic> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\"><mml:mi>v</mml:mi></mml:math> with respect to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M11\"><mml:mi>W</mml:mi></mml:math>. The set <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M12\"><mml:mi>W</mml:mi></mml:math> is a <jats:italic>multiresolving set</jats:italic> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M13\"><mml:mi>G</mml:mi></mml:math> if the multirepresentations of every two distinct vertices of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M14\"><mml:mi>G</mml:mi></mml:math> with respect to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M15\"><mml:mi>W</mml:mi></mml:math> are distinct. The multiresolving set of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M16\"><mml:mi>G</mml:mi></mml:math> having the minimum cardinality is called a <jats:italic>multibasis</jats:italic> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M17\"><mml:mi>G</mml:mi></mml:math>. The cardinality of a multibasis of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M18\"><mml:mi>G</mml:mi></mml:math> is the <jats:italic>multidimension</jats:italic><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M19\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">dim</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:mfenced></mml:math> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M20\"><mml:mi>G</mml:mi></mml:math>. A caterpillar <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M21\"><mml:mtext>ca</mml:mtext><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math> is called a <jats:italic>symmetric caterpillar</jats:italic> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M22\"><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> for all integers <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M23\"><mml:mi>i</mml:mi></mml:math> with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" id=\"M24\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>i</mml:mi><mml:mo>≤</mml:mo><mml:mi>s</mml:mi></mml:math>. In this work, the multiresolving sets of symmetric caterpillars are studied.</jats:p>

/pdf/the-multibases-of-symmetric-caterpillars-579p89ompv.pdf

The Multibases of Symmetric Caterpillars

Let v be a vertex of a connected graph G , and let W = { w 1 , w 2 , ..., w k } be a set of vertices of G . The multirepresentation of v with respect to W is the k -multiset mr ( v | W ) = { d ( v,w 1 ) ,d ( v,w 2 ) ,...,d ( v,w k ) } . A set W is called a multiresolving set of G if no two vertices of G have the same multirepresentations with respect to W . The multidimension of G is the minimum cardinality of a multiresolving set of G . In this paper, we characterize the caterpillars with multidimension 3.

The Characterization of Caterpillars with Multidimension 3

Suppose the set W = {s1, s2,…, sk} is a subset of the vertex set V(G). The representation of a vertex v of G with respect to W as follows rm(v|W) = { d(v, s1), d(v, s2), … , d(v, sk)} where d(v, si),1 ≤ i ≤ k is the distance between the vertex v with the vertices of set W together with their multiplicities. The set W is called the m-local resolving set of G if every two adjacent vertices of G have distinct representation with respect to W. If G has an m-local resolving set, then an m-local resolving set having minimum cardinality is called a local multiset basis and its cardinality is called the local multiset dimension of G, denoted by mdl (G). We say that G has an infinite local multiset dimension and we write mdl(G) = ∞. In this paper, we determine the local multiset dimension of related cycle graphs namely kayak paddles graph, and cycles with chord. 

Local multiset dimension of related cycle graphs

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B⊆V(Γ). If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions.

/pdf/a-new-technique-to-uniquely-identify-the-edges-of-a-graph-1518k242.pdf

A New Technique to Uniquely Identify the Edges of a Graph

For a set of vertices and a vertex of a connected graph , the multirepresentation of with respect to is the -multiset where is the distance between the vertices and for . The set is a multiresolving set of if every two distinct vertices of have distinct multirepresentations with respect to . The minimum cardinality of a multiresolving set of is the multidimension of . It is shown that, for every pair of integers with and , there is a connected graph of order with . For a multiset and an integer , we define . A multisimilar equivalence relation on with respect to is defined by if for some integer . We study the relationship between the elements in multirepresentations of vertices that belong to the same multisimilar equivalence class and also establish the upper bound for the cardinality of a multisimilar equivalence class. Moreover, a multiresolving set with prescribed multisimilar equivalence classes is presented.

/pdf/the-multiresolving-sets-of-graphs-with-prescribed-41uoc2uyau.pdf

The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes

For an ordered set W = {w1,w2, ...,wk} of k distinct vertices in a connected graph G, the representation of a vertex v of G with respect to W is the k-vector r(v|W) = (d(v,w1), d(v,w2), ..., d(v,wk)), where d(v,wi) is the distance from v to wi for 1 ≤ i ≤ k. The setW is called a connected local resolving set of G if the representations of every two adjacent vertices of G with respect to W are distinct and the subgraph ⟨W⟩ induced by W is connected. A connected local resolving set of G of minimum cardinality is a connected local basis of G. The connected local dimension cld(G) of G is the cardinality of a connected local basis of G. In this paper, the connected local dimensions of some well-known graphs are determined. We study the relationship between connected local bases and local bases in a connected graph, and also present some realization results.

Supachoke Isariyapalakul

Papers

The Multiresolving Sets of Graphs with Prescribed Multisimilar Equivalence Classes

The Multibases of Symmetric Caterpillars

The Characterization of Caterpillars with Multidimension 3

Graph Realizations Constrained by Connected Local Dimensions and Connected Local Bases

Bounds on the connected local dimension of graphs in terms of the marked dimension and the clique number