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In particular, our algorithmic applications include single-exponential (or slightly super-exponential) algorithms for well-studied problems such as MIN BISECTION, STEINER CUT, s-WAY CUT, and EDGE MULTIWAY CUT-UNCUT on UDGs these algorithms are substantially faster than the best known algorithms for these problems on general graphs. Having this decomposition at hand, we show that the design of parameterized algorithms for some cut problems becomes elementary. For any bag in the tree decomposition and any edge in S with both endpoints in the bag, either its endpoints lie in different pieces or they lie in a piece which is a clique. Moreover, for any subset S of at most k edges in G, there is a tree decomposition in the collection such that S is well preserved in the decomposition in the following sense. Each bag in any of these tree decompositions can be partitioned into O(k) connected pieces (we call this measure the chunkiness of the tree decomposition). In gen- eral, this problem is known to be R-complete. unit disk graph recognition, is an important geometric problem, and has many application areas. Deciding whether there exists an embed- ding of a given unit disk graph, i.e. We prove that there is an algorithm that given a UDG G and a positive integer k, runs in polynomial time and outputs a collection of O(k) tree decompositions of G with the following properties. Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Second, we present a "parameteric version" of our new decomposition theorem. #Unit disk graph freeAdditionally, this result answers an open question posed by Hajiaghayi (regarding the existence of contraction decompositions for classes of graphs beyond H-minor free graphs. Even bounded-degree UDGs can contain arbitrarily large cliques as minors, therefore our result is a significant advance in the study of contraction decompositions. Prior studies on this topic exhibited that the classes of planar graphs, graphs of bounded genus and H-minor free graphs admit a Contraction Decomposition Theorem. First, our new decomposition theorem shows that the class of UDGs admits a Contraction Decomposition Theorem. We give a new decomposition theorem in unit disk graphs (UDGs) and demonstrate its applicability in the fields of Structural Graph Theory and Parameterized Complexity. As she later finds out, the sword is actually one of a hundred swords collectively known as the Swords of Vaul, after the Eldar god that created them.
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