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A connected dominating set is a dominating set that is also connected.If S is a connected dominating set, one can form a spanning tree of G in which S forms the set of non-leaf vertices of the tree; conversely, if T is any spanning tree in a graph with more than two vertices, the non-leaf vertices of T form a connected dominating set.
A connected dominating set of a graph G is a set D of vertices with two properties: Any node in D can reach any other node in D by a path that stays entirely within D. That is, D induces a connected subgraph of G. Every vertex in G either belongs to D or is adjacent to a vertex in D. That is, D is a dominating set of G.
An edge dominating set is also known as a line dominating set. Figures (a)–(d) are examples of edge dominating sets (thick red lines). A minimum edge dominating set is a smallest edge dominating set. Figures (a) and (b) are examples of minimum edge dominating sets (it can be checked that there is no edge dominating set of size 2 for this graph).
Dominating set, a.k.a. domination number [3]: GT2 NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf spanning tree problem. [3]: ND2 Feedback vertex set [2] [3]: GT7
Alternatively, (1) a maximal independent set is a dominating set, and (2) the complement of a maximal independent set is also a dominating set if there are no isolated vertices. The figure on the right shows a weak 2-coloring, which is also a domatic partition of size 2: the dark nodes are a dominating set, and the light nodes are another ...
The dominance frontier of a node d is the set of all nodes n i such that d dominates an immediate predecessor of n i, but d does not strictly dominate n i. It is the set of nodes where d 's dominance stops. A dominator tree is a tree where each node's children are those nodes it immediately dominates. The start node is the root of the tree.
A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. The top two P 3 graphs are maximal independent sets while the bottom two are independent sets, but not maximal. The maximum independent set is represented by the top left.
Dominating set is the problem of selecting a set of vertices (the dominating set) in a graph such that all other vertices are adjacent to at least one vertex in the dominating set. The Dominating set problem was shown to be NP complete through a reduction from Set cover.