Abstract In this work, we study the following problem: given a connected graph G , can we reduce the domination number of G by at least one using k edge contractions, for some fixed integer k > 0 ? We show that for k = 1 (resp. k = 2 ), the problem is NP -hard (resp. coNP -hard). We further prove that for k = 1 , the problem is W [1]-hard parameterized by domination number plus the mim-width of the input graph, and that it remains NP -hard when restricted to chordal { P 6 , P 4 + P 2 } -free graphs, bipartite graphs and { C 3 , … , C l } -free graphs for any l ≥ 3 . We also show that for k = 1 , the problem is coNP -hard on subcubic claw-free graphs, subcubic planar graphs and on 2 P 3 -free graphs. On the positive side, we show that for any k ≥ 1 , the problem is polynomial-time solvable on ( P 5 + p K 1 ) -free graphs for any p ≥ 0 and that it can be solved in FPT -time and XP -time when parameterized by treewidth and mim-width, respectively. Finally, we start the study of the problem of reducing the domination number of a graph via vertex deletions and edge additions and, in this case, present a complexity dichotomy on H -free graphs.

中文翻译：

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通过边收缩和顶点删除减少图的支配数

摘要 在这项工作中，我们研究了以下问题：给定一个连通图 G，对于某个固定整数 k > 0，我们能否使用 k 边收缩将 G 的支配数减少至少一个？我们表明，对于 k = 1 (resp. k = 2 )，问题是 NP -hard (resp. coNP -hard)。我们进一步证明，对于 k = 1 ，问题是 W [1]-hard 参数化的控制数加上输入图的 mim-width，并且当限制为 chordal { P 6 , P 4 + 时它仍然是 NP-hard P 2 } -free 图、二部图和 { C 3 , … , C l } -free 图对于任何 l ≥ 3 。我们还表明，对于 k = 1 ，问题在亚立方无爪图、亚立方平面图和 2 P 3 无图上是 coNP -hard 问题。从积极的方面来说，我们表明对于任何 k ≥ 1 ，对于任何 p ≥ 0，该问题在 ( P 5 + p K 1 ) 自由图上是多项式时间可解的，并且当分别由 treewidth 和 mim-width 参数化时，它可以在 FPT 时间和 XP 时间中解决。最后，我们开始研究通过顶点删除和边添加来减少图的支配数的问题，在这种情况下，在无 H 图上提出复杂性二分法。