We consider the complexity of recognizing k-clique-extendible graphs (k-C-E graphs) introduced by Spinrad (Efficient Graph Representations, AMS 2003), which are generalizations of comparability graphs. A graph is k-clique-extendible if there is an ordering of the vertices such that whenever two overlapping k-cliques A and B have \(k-1\) common vertices, and these common vertices appear between the two vertices \(a,b\in (A{\setminus } B)\cup (B{\setminus } A)\) in the ordering, there is an edge between a and b, implying that \(A\cup B\) is a \((k+1)\)-clique. Such an ordering is said to be a k-C-E ordering. These graphs arise in applications related to modelling preference relations. Recently, it has been shown that a maximum clique in such a graph can be found in \(n^{O(k)}\) time [Hamburger et al. 2017] when the ordering is given. When k is 2, such graphs are precisely the well-known class of comparability graphs and when k is 3 they are called triangle-extendible graphs. It has been shown that triangle-extendible graphs appear as induced subgraphs of visibility graphs of simple polygons, and the complexity of recognizing them has been mentioned as an open problem in the literature. While comparability graphs (i.e. 2-C-E graphs) can be recognized in polynomial time, we show that recognizing k-C-E graphs is NP-hard for any fixed \(k \ge 3\) and co-NP-hard when k is part of the input. While our NP-hardness reduction for \(k \ge 4\) is from the betweenness problem, for \(k=3\), our reduction is an intricate one from the 3-colouring problem. We also show that the problems of determining whether a given ordering of the vertices of a graph is a k-C-E ordering, and that of finding a maximum clique in a k-C-E graph, given a k-C-E ordering, are hard for the parameterized complexity classes co-W[1] and W[1] respectively, when parameterized by k. However we show that the former is fixed-parameter tractable when parameterized by the treewidth of the graph. We also show that the dual parameterizations of all the problems that we study are fixed parameter tractable.

我们考虑了识别由 Spinrad（Efficient Graph Representations，AMS 2003）引入的k -clique-extendible 图（k -CE图）的复杂性，它们是可比性图的概括。一个图是k -clique-extendible，如果顶点的排序使得每当两个重叠的k- cliques A和B具有\(k-1\) 个公共顶点，并且这些公共顶点出现在两个顶点之间\(a ,b\in (A{\setminus } B)\cup (B{\setminus } A)\)在排序中，a和b之间存在边，暗示\(A\cup B\)是一个\((k+1)\) -集团。这种排序被称为k- CE 排序。这些图出现在与建模偏好关系相关的应用程序中。最近，已经证明可以在\(n^{O(k)}\)时间内找到此类图中的最大集团[Hamburger et al. 2017] 下订单时。当k为 2 时，此类图正是众所周知的可比性图类，当k是 3 它们被称为三角形可扩展图。已经表明，三角形可扩展图作为简单多边形的可见性图的诱导子图出现，并且识别它们的复杂性在文献中被提及为一个未解决的问题。虽然可以在多项式时间内识别可比性图（即 2-CE 图），但我们表明，识别k -CE 图对于任何固定的\(k \ge 3\)和co-NP -hard 当k是一部分时是 NP-hard的输入。虽然我们对\(k \ge 4\)的 NP 硬度降低来自介数问题，但对于\(k=3\)，我们的减少是来自 3-coloring 问题的一个复杂的减少。我们还表明，确定图顶点的给定排序是否是k -CE 排序的问题，以及在给定k -CE 排序的情况下，在k -CE 图中找到最大团的问题对于当参数化为 k时，参数化复杂度类分别为co-W[1]和W[1]。然而，我们表明，当由图的树宽参数化时，前者是固定参数可处理的。我们还表明，我们研究的所有问题的双参数化都是固定参数易处理的。