A set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality for the input graph $G$. Given a positive integer $k$ and a graph $G = (V, E)$, the Outer-connected Domination Decision problem is to decide whether $G$ has an outer-connected dominating set of cardinality at most $k$. The Outer-connected Domination Decision problem is known to be NP-complete, even for bipartite graphs. We study the problem of outer-connected domination on sparse graphs from the perspective of parameterized complexity and show that it is W[1]-hard on d-degenerate graphs, while the original connected dominating set has FTP algorithm on d-degenerate graphs.
翻译：如果每个不以美元计的顶点都与至少一个顶点相邻($$),且导引的基点为$V\setminus D$的子图是相连的,则一套美元($D ) = G = (V,E) = 美元,则称为外联的支配 $G = ($G) = (V,E) 。如果每个顶点不以美元计的美元与至少一个顶点相邻,且导导引的基点为$V \setminus D$ 。最小外联的定点问题是要找到一组外联的最小主点。鉴于正正整值美元和G = (V,E), 外联的定点是 $G$ 是否具有外联的顶点, 而原始的顶点定是图形的顶点为 。 我们从参数化的复杂度角度研究微点图形的外部控制问题, 并显示它在 d-degenate 图形上是 W[1]- hard 。
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