Not every graph can be reconstructed from its boundary distance matrix

Cáceres, José; Pelayo, Ignacio M

doi:10.22049/cco.2026.30664.2580

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	Not every graph can be reconstructed from its boundary distance matrix
Communications in Combinatorics and Optimization
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 06 اردیبهشت 1405 اصل مقاله (728.18 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2026.30664.2580
نویسندگان
José Cáceres^* ¹؛ Ignacio M Pelayo²
¹CDTIME and Departamento de Matemáticas, Universidad de Almería, Almería, Spain
²Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
چکیده
A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary vertices. The boundary distance matrix $\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $\kappa$, being $\kappa$ the order of $\partial(G)$, such that for every $i,j\in \partial(G)$, $[\hat{D}_G]_{ij}=d_G(i,j)$. In a recent paper [doi.org/10.7151/dmgt.2567], it was shown that if a graph $G$ is either a block graph or a unicyclic graph, then $G$ is uniquely determined by the boundary distance matrix $\hat{D}_{G}$ of $G$, and it was also conjectured that this statement holds for every connected graph $G$, whenever both the order $n$ and the $\|\partial(G)\|$ (and thus also the boundary distance matrix) of $G$ are prefixed (i.e., the set of boundary vertices is known as a subset of $V$, but not their identities). After proving that this conjecture is true for several graph families, such as being of diameter 2, having order at most $n=6$ or being Ptolemaic, we show that this statement does not hold when considering, for example, either the family of split graphs of diameter 3 and order at least $n=10$ or the family of distance-hereditary graphs of order at least $n=8$.
کلیدواژه‌ها
Boundary؛ Distance Matrix؛ Ptolemaic graphs؛ Interval graphs

مراجع
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Not every graph can be reconstructed from its boundary distance matrix