论文标题
极端混合度量尺寸相对于循环数
Extremal mixed metric dimension with respect to the cyclomatic number
论文作者
论文摘要
在图G中，区分v（g）u e（g）每个元素的最小有序的顶点的基数称为G的混合度量尺寸，并用mdim（g）表示。 In [12] it was conjectured that for a graph G with cyclomatic number c(G) it holds that mdim(G) <= L1(G) + 2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree >= 3. In this paper we determine that for every Theta Graph G，混合度量尺寸MDIM（G）等于3或4，当G是平衡的Theta图时，将达到4。因此，对于平衡的theta图，上述不等式也很紧。我们通过进一步猜想的是，除了这里提到的图外，我们还没有其他图表结束了论文。
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)U E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that for a graph G with cyclomatic number c(G) it holds that mdim(G) <= L1(G) + 2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree >= 3. In this paper we determine that for every Theta graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Theta graph. Thus, for balanced Theta graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G) = L1(G) + 2c(G) holds.
