完全頂點覆蓋是一種頂點覆蓋，其誘導子圖的每個連通子圖包含至少兩個點。t-完全頂點覆蓋是一種完全頂點覆蓋，其誘導子圖的每個連通子圖包含至少t 個點。完全頂點覆蓋問題的解答是最小的完全頂點覆蓋，而t-完全頂點覆蓋問題的解答是最小的t-完全頂點覆蓋。在這篇論文中，首先我們證明t-完全頂點覆蓋問題在周長任意大的次3-正則網格連通圖上是NP 完全，接著我們證明t-完全頂點覆蓋問題在3-正則3-連通平面圖上是NP 完全，最後我們證明t-完全頂點覆蓋問題在周長任意大的次3-正則連通圖上是APX 完全。

A total vertex cover is a vertex cover whose induced subgraph consists of a set of connected components, each of which contains at least two vertices. A t-total vertex cover is a total vertex cover where each component of its induced subgraph contains at least t vertices. The total vertex cover (TVC) problem and the t-total vertex cover (t-TVC) problem ask for the corresponding cover set with minimum cardinality, respectively. In this paper, we first show that the t-TVC problem is NP-complete for connected subcubic grid graphs of arbitrary large girth. Next, we show that the t-TVC problem is NP-complete for 3-connected cubic planar graphs. Moreover, we show that the t-TVC problem is APX-complete for connected subcubic graphs of arbitrary large girth.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 NP-completeness for subcubic grid graphs of arbitrary girth 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 t-TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 NP-completeness for 3-connected cubic planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 t-TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Approximation hardness for connected subcubic graphs of arbitrary girth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 t-TVC problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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