Abstract
Many important graph classes, such as interval graphs, comparability graphs and AT-free graphs, show some kind of linear structure. In this paper we try to capture the notion of linearity and show some algorithmic implications. In the first section we discuss the notion of linearity of graphs and give some motivation for its usefulness for particular graph classes. The second section deals with the knotting graph, a combinatorial structure that was defined by Gallai long ago and that has various nice properties with respect to our notion of linearity. Next we define intervals of graphs in Sect. 3. This concept generalizes betweenness in graphs—a crucial notion for capturing linear structure in graphs. In the last section we give a practical example of how to use the linear structure of graphs algorithmically. In particular we show how to use these structural insights for finding maximum independent sets in AT-free graphs in \(O(n\overline{m})\) time, where \(\overline{m}\) denotes the number of non-edges of the graph G.
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