The main result in this paper is a new lower bound to the eternal vertex cover number (evc number) of an arbitrary graph G in terms of the size of the smallest vertex cover in G that includes all the cut vertices of G. As a consequence, we obtain a quadratic complexity algorithm for finding the evc number of any chordal graph. Another consequence is a polynomial time approximation scheme for finding the evc number of internally triangulated planar graphs, for which exact determination of evc number is known to be NP-hard (Babu et al. in Discrete Appl Math, 2021. https://doi.org/10.1016/j.dam.2021.02.004). The lower bound is proven by considering a decomposition of the graph into a collection of edge disjoint induced subgraphs, and deriving a lower bound for the evc number of the whole graph in terms of bounds obtained for the subgraphs. As another consequence of the bounding technique, we obtain a construction of a family of biconnected bipartite graphs such that for any \(\epsilon >0\), there exists a graph in the family such that the ratio of its evc number to the size of its minimum vertex cover exceeds \(2-\epsilon \). This construction is asymptotically optimal, as it is known (Klostermeyer and Mynhardt in Aust J Comb 45:235–250, 2009) that this ratio has to be strictly less than 2 for biconnected graphs.

本文的主要结果是根据G中包含G 的所有切割顶点的G 中最小顶点覆盖的大小，得到任意图G的永恒顶点覆盖数（evc 数）的新下界. 因此，我们获得了用于查找任何弦图的 evc 数的二次复杂度算法。另一个结果是多项式时间近似方案，用于查找内部三角平面图的 evc 数，已知其 evc 数的精确确定是 NP 难的（Babu 等人在 Discrete Appl Math, 2021. https://doi .org/10.1016/j.dam.2021.02.004）。通过考虑将图分解为边不相交的诱导子图的集合，并根据为子图获得的界限推导出整个图的 evc 数的下界，来证明下界。作为边界技术的另一个结果，我们获得了一个双连通二部图族的构造，使得对于任何\(\epsilon >0\)，族中存在一个图，使得其 evc 数与其最小顶点覆盖的大小之比超过\(2-\epsilon \)。这种构造是渐近最优的，众所周知（Klostermeyer 和 Mynhardt in Aust J Comb 45:235–250, 2009）对于双连通图，这个比率必须严格小于 2。