Minimizing the number of edges in (Pk ∪ K3)-saturated connected graphs

Yuying Li, Kexiang Xu
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Abstract

For a graph H, a graph G is H-saturated if it contains no copy of H as a (not necessarily induced) subgraph, but the addition of any edge missing from G creates a copy of H in the resultant graph. The connected saturation number sat'(n,H) is defined as the minimum number of edges in H-saturated connected graphs on n vertices. In this paper we consider the (Pk∪K3)-saturated connected graphs on n vertices and focus on the determination of  sat'(n,Pk∪K3). We prove that n+2≤sat'(n,Pk∪ K3)≤n+(3k-6)/2 for n>(3k+6)/2 with k ≥ 4 and characterize the extremal graphs at which the upper bounds are attained. Moreover, the exact values of sat'(n,Pk∪ K3) are determined with k ∈{2,3,4}.
最小化(Pk∪K3)饱和连通图中的边数
对于图H,如果图G不包含H的副本(不一定是诱导的)子图,则图G是H饱和的,但是添加G中缺失的任何边会在生成的图中创建H的副本。连通饱和数sat'(n,H)定义为n个顶点上H饱和连通图的最小边数。本文考虑n个顶点上的(Pk∪K3)饱和连通图,重点讨论了sat'(n,Pk∪K3)的确定问题。证明了当k≥4时,当n>(3k+6)/2时,n+2≤sat'(n,Pk∪K3)≤n+(3k-6)/2,并刻画了极值图的上界。此外,sat'(n,Pk∪K3)的确切值由k∈{2,3,4}确定。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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