论文标题
边缘度量尺寸通过分层产品和整数线性编程
Edge metric dimensions via hierarchical product and integer linear programming
论文作者
论文摘要
如果$ s = \ {v_1，\ ldots，v_k \} $是连接图$ g $和$ e $的顶点的订购子集，则是$ g $的边缘，则vector $ r_g（e | s）=（d_g（v_1，e），v_1，e），\ ldots，\ ldots，d_g（eff _ v_kkk kk，e） $ e $。如果$ g $的顶点具有成对的不同边缘度量$ s $代表，那么$ s $是$ g $的边缘度量生成器。最小的边缘公制生成器的基数是$ g $的边缘度量尺寸$ {\ rm edim}（g）$。证明了分层产品$ g（u）\ sqcap h $的边缘度量尺寸的一般锋利的上限。当$ | u |时，确切的公式是派生的。 = 1 $。提出了用于计算边缘度量维度的整数线性编程模型。提供了几个示例，这些示例证明了如何应用这两种方法来获得某些适用图的边缘度量尺寸。
If $S=\{v_1,\ldots, v_k\}$ is an ordered subset of vertices of a connected graph $G$ and $e$ is an edge of $G$, then the vector $r_G(e|S) = (d_G(v_1,e), \ldots, d_G(v_k,e))$ is the edge metric $S$-representation of $e$. If the vertices of $G$ have pairwise different edge metric $S$-representations, then $S$ is an edge metric generator for $G$. The cardinality of a smallest edge metric generator is the edge metric dimension ${\rm edim}(G)$ of $G$. A general sharp upper bound on the edge metric dimension of hierarchical products $G(U)\sqcap H$ is proved. Exact formula is derived for the case when $|U| = 1$. An integer linear programming model for computing the edge metric dimension is proposed. Several examples are provided which demonstrate how these two methods can be applied to obtain the edge metric dimensions of some applicable graphs.
