Triangle graphs and their coloring
In this paper, we present results on two subclasses of trapezoid graphs, including simple trapezoid graphs and triangle graphs (also known as PI graph in ). Simple trapezoid graphs and triangle graphs are proper subclasses of trapezoid graphs [5, 3]. Here we show that simple trapezoid graphs and triangle graphs are also two distinct subclasses of trapezoid graphs.
Further, given the triangle representation and assuming that upper vertices of the triangles are listed in sorted order, we show that optimization problems on triangle graphs including finding the maximum independent set and minimum clique partition can be found in O(n log α) time where α is the size of the largest independent set. The maximum clique and minimum vertex coloring in triangle graphs can be found in O(n log χ) where χ is the chromatic number of the underlying triangle graph.
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