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Algorithms and obstructions for linear-width and related search parameters. (English) Zbl 0958.05124
Define the width of a linear ordering $$(e_1, \ldots, e_r)$$ of the edges of a graph $$G=(V,E)$$ to be the maximum, over all $$i$$, of the number of vertices that are incident with edges from $$\{e_1, \ldots, e_i\}$$, and are incident with edges from $$\{e_{i+1}, \ldots, e_r\}$$. The linear-width of a graph is the minimum width over all orderings of its edges. If we take a minor of a graph, then the linear-width cannot increase. Thus, for each $$k$$, the class of graphs of linear-width at most $$k$$ is closed under taking of minors, and hence has, by the work of Robertson and Seymour, a finite characterization by its obstruction set. In this paper, the obstruction set for linear-width at most two is given: it contains exactly 57 graphs. Also, with help of a connection to the mixed search number, the paper identifies the set of acyclic forbidden minors in the obstruction set of linear-width at most $$k$$, for any $$k\geq 1$$; and linear time algorithms for checking linear-width, mixed search, or edge-search at most two are given; these also can construct the corresponding edge orderings or search strategies.

##### MSC:
 05C83 Graph minors 05C85 Graph algorithms (graph-theoretic aspects) 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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