
On the Maximum Cardinality Cut Problem in Proper Interval Graphs and Related Graph Classes
Although it has been claimed in two different papers that the maximum ca...
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Assistance and Interdiction Problems on Interval Graphs
We introduce a novel framework of graph modifications specific to interv...
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Online Coloring of Short Intervals
We study the online graph coloring problem restricted to the intersectio...
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ConflictFree Coloring: Graphs of Bounded Clique Width and Intersection Graphs
Given an undirected graph, a conflictfree coloring (CFON*) is an assign...
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Coloring evenholefree graphs with no star cutset
A hole is a chordless cycle of length at least 4. A graph is evenholef...
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EPTAS for Max Clique on Disks and Unit Balls
We propose a polynomialtime algorithm which takes as input a finite set...
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On tradeoffs between width and filllike graph parameters
In this work we consider two twocriteria optimization problems: given a...
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UBubble Model for Mixed Unit Interval Graphs and its Applications: The MaxCut Problem Revisited
Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs; a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semiclosed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not clawfree, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs. The original bubble model was used by Boyaci, Ekim, and Shalom for proving the polynomiality of the MaxCut problem on unit interval graphs. However, we found a significant mistake in the proof which seems to be hardly repairable. Moreover, we demonstrate the advantages of such a model by providing a subexponentialtime algorithm solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model. In addition, it gives us a polynomialtime algorithm for specific mixed unit interval graphs; that improves a stateoftheart result even for unit interval graphs. We further provide a better algorithmic upperbound on the cliquewidth of mixed unit interval graphs. Cliquewidth is one of the most general structural graph parameters, where a large group of natural problems is still solvable in the tracktable time when an efficient representation is given. Unfortunately, the exact computation of the cliquewidth representation is NPhard. Therefore, good upperbounds on cliquewidth are highly appreciated.
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