In a graph $G,$ the cardinality of the smallest ordered set of vertices that distinguishes every element of $V\left(G\right)\cup E\left(G\right)$ is called the mixed metric dimension of $G,$ and it is denoted by $\mathrm{mdim}\left(G\right).$ In [12] it was conjectured that every graph $G$ with cyclomatic number $c\left(G\right)$ satisfies $\mathrm{mdim}\left(G\right)\le L_1\left(G\right)+2c\left(G\right)$ where $L_1\left(G\right)$ 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 $\ge 3$. In this paper we determine that for every $\Theta$-graph $G,$ the mixed metric dimension $\mathrm{mdim}\left(G\right)$ 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 $\mathrm{mdim}\left(G\right)=L_1\left(G\right)+2c\left(G\right)$ holds.

在图中 $G，$ 区分顶点的每个元素的最小有序顶点集的基数 $\mathrm{伏特}（G）\cup E（G）$ 称为的混合度量维 $G，$ 它由表示 $\mathrm{mdim}（G）。$ 在[12]中推测每个图 $G$ 带圈数 $C（G）$ 满足 $\mathrm{mdim}（G）\le {\mathrm{大号}}_{\mathrm{1个}}（G）+\mathrm{2个}C（G）$ 在哪里 ${\mathrm{大号}}_{\mathrm{1个}}（G）$ 是中的叶子数 $G$。已经证明，等式适用于所有树，更普遍地适用于具有边不相交周期的图，其中每个周期恰好具有一个度数顶点$\ge 3$。在本文中，我们确定$\Theta$-图形 $G，$ 混合指标维度 $\mathrm{mdim}（G）$ 等于3或4，且仅当且仅当达到4 $G$ 是平衡的 $\Theta$-图形。因此，为了平衡$\Theta$图上面的不平等也很严密。通过进一步推测，本文得出结论，除此处提到的图形外，没有其他图形具有相等性。$\mathrm{mdim}（G）={\mathrm{大号}}_{\mathrm{1个}}（G）+\mathrm{2个}C（G）$ 持有。