International Journal of Computing Science and Applied Mathematics-IJCSAM
Vol. 11 No. 2 (2025)

Partition Dimension of Bridge Graphs Between Complete and Star Graphs

Amrullah Amrullah (Mataran University)
Laila Hayati (Universitas Mataram)
Junaidi Junaidi (Universitas Mataram)



Article Info

Publish Date
15 Dec 2025

Abstract

This paper investigates the determination of the partition dimension for a \emph{bridge graph} formed by connecting a clique $K_n$ and a star $K_{1,m}$ through a single edge. Although the partition dimension has been extensively studied for various families and graph operations, the mixed dense--sparse case on $B(K_n,K_{1,m})$ remains unsettled, since the result is sensitive to the position of the bridge edge and the balance between the size parameters $n$ and $m$.We combine distance symmetry arguments, leaf-based constraints at the star center, and explicit constructions of distinguishing partitions to obtain tight values of the partition dimension. The study begins with the basic cases $K_1$ and $K_2$, and then proceeds to the general case with parameters $n\ge 2$. The main result shows that for the \emph{central bridge} ($e=v_1x$), it holds that $pd(B)=n-1$ if $m<n$, $pd(B)=n$ if $m=n$, and $pd(B)=m$ if $m>n$; for the \emph{leaf bridge} ($e=v_1u_1$), it holds that $pd(B)=n$ when $m\le n$, and$pd(B)=m-1$ when $m>n$. These results demonstrate that the location of the bridge edge, together with the size parameters $m$ and $n$ of the components, can sharpen the partition dimension value of the graph prior to the bridging operation.

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Journal Info

Abbrev

ijcsam

Publisher

Subject

Mathematics

Description

IJCSAM (International Journal of Computing Science and Applied Mathematics) is an open access journal publishing advanced results in the fields of computations, science and applied mathematics, as mentioned explicitly in the scope of the journal. The journal is geared towards dissemination of ...