## A unified approach to domination problems on interval graphs.(English)Zbl 0658.05040

A subset D of the vertex set V(G) of a graph G is called dominating (or total dominating) in G, if for each vertex $$x\in V(G)-D$$ (or for each vertex $$x\in V(G)$$ respectively) there exists a vertex $$y\in D$$ adjacent to x. A dominating set of G which induces a connected subgraph of G is called connected. Also independent dominating sets are considered. The mentioned concepts are studied for interval graphs, i.e. intersection graphs of families of intervals on a real line. Some recurrence relations for various types of dominating sets in interval graphs are stated. These relations lead to linear-time algorithms for the weighted version of the various dominating problems in a unified way.

### MSC:

 05C35 Extremal problems in graph theory 05C99 Graph theory 68R10 Graph theory (including graph drawing) in computer science

### Keywords:

dominating set; interval graphs
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### References:

  Bertossi, A.A., Total domination in interval graphs, Inform. process. lett., 23, 131-134, (1986) · Zbl 0604.05032  Booth, K.S.; Johnson, J.H., Dominating sets in chordal graphs, SIAM J. comput., 11, 191-199, (1982) · Zbl 0485.05055  Booth, K.S.; Lueker, G.S., Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms, J. comput. system sci., 13, 335-379, (1976) · Zbl 0367.68034  Farber, M., Domination, independent domination, and duality in strongly chordal graphs, Discrete appl. math., 7, 115-130, (1984) · Zbl 0531.05045  Garey, M.R.; Johnson, D.S., Computers and intractability: A guide to the theory of NP-completeness, (1979), Freeman San Francisco, CA · Zbl 0411.68039  Keil, J.M., Total domination in interval graphs, Inform. process. lett., 22, 171-174, (1986) · Zbl 0595.05063  Laskar, R.; Pfaff, J.; Hedetniemi, S.M.; Hedetniemi, S.T., On the algorithmic complexity of total domination, SIAM J. alg. disc. meth., 5, 420-425, (1984) · Zbl 0576.68056  White, K.; Farber, M.; Pulleyblank, W., Steiner trees, connected domination and strongly chordal graphs, Networks, 15, 109-124, (1985) · Zbl 0579.05050
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