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... and bounded-treewidth graphs. A key property of bounded arboricity graphs that has been exploited in various algorithmic applications is the following Nash-Williams Theorem. Theorem 1 (Nash-Williams =-=[11,12]-=-). A graph G = (V, E) has arboricity α(G) if and only if α(G) > 0 is the smallest number of sets E1, . . . , E α(G) that E can be partitioned into, such that each subgraph (V, Ei) is a forest. This th...

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... and bounded-treewidth graphs. A key property of bounded arboricity graphs that has been exploited in various algorithmic applications is the following Nash-Williams Theorem. Theorem 1 (Nash-Williams =-=[11,12]-=-). A graph G = (V, E) has arboricity α(G) if and only if α(G) > 0 is the smallest number of sets E1, . . . , E α(G) that E can be partitioned into, such that each subgraph (V, Ei) is a forest. This th...

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...g” [1], where orientations are used to develop more efficient algorithms for finding simple cycles and paths. Another fundamental example is in data structures for quickly answering adjacency queries =-=[2,3,4]-=-, where a corientation of a (dynamic) graph G is used to answer adjacency queries in O(c) time using only linear space. These techniques [2,3,4] were further generalized to answer short-path queries [...

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...ry vertex is at most 1, hence the union of the oriented forests has out-degree bounded by α(G). There exists a polynomial-time algorithm that computes for a (static) graph G the exact arboricity α(G) =-=[13]-=-, and a linear-time algorithm that computes a (2α(G) − 1)-orientation [14]. For every (static) graph G, the minimum-possible maximum out-degree is closely related to α(G): the argument above provides ...

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... each vertex is at most 1, hence in the union of the oriented forests the out-degree of each vertex is at most α(G). There 2 exists a polynomial-time algorithm for computing the exact arboricity α(G) =-=[GW92]-=-, and a linear-time algorithm for computing a (2α(G) − 1)-orientation for a static graph G [AMZ97]. For every graph G, the maximum out-degree (of its edge orientations) is closely related to α(G): The...

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...ng only linear space. These techniques [2,3,4] were further generalized to answer short-path queries [5]. Additional examples for the algorithmic use of low-degree orientations include load balancing =-=[6]-=-, maximal matchings [7], counting subgraphs in sparse graphs [8], prize-collecting TSPs and Steiner Trees [9], reporting all maximal independent sets [10], answering dominance queries [10], subgraph l...

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...low-degree orientations include load balancing [6], maximal matchings [7], counting subgraphs in sparse graphs [8], prize-collecting TSPs and Steiner Trees [9], reporting all maximal independent sets =-=[10]-=-, answering dominance queries [10], subgraph listing problems (listing triangles and 4-cliques) in planar graphs [2], and computing the girth [5]. Efficient Data Communication. The efficiency of netwo...

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...g” [1], where orientations are used to develop more efficient algorithms for finding simple cycles and paths. Another fundamental example is in data structures for quickly answering adjacency queries =-=[2,3,4]-=-, where a corientation of a (dynamic) graph G is used to answer adjacency queries in O(c) time using only linear space. These techniques [2,3,4] were further generalized to answer short-path queries [...

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...gree bounded by α(G). There exists a polynomial-time algorithm that computes for a (static) graph G the exact arboricity α(G) [13], and a linear-time algorithm that computes a (2α(G) − 1)-orientation =-=[14]-=-. For every (static) graph G, the minimum-possible maximum out-degree is closely related to α(G): the argument above provides an orientation with maximum out-degree at most α(G), but the maximum out-d...

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...hese techniques [2,3,4] were further generalized to answer short-path queries [5]. Additional examples for the algorithmic use of low-degree orientations include load balancing [6], maximal matchings =-=[7]-=-, counting subgraphs in sparse graphs [8], prize-collecting TSPs and Steiner Trees [9], reporting all maximal independent sets [10], answering dominance queries [10], subgraph listing problems (listin...

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...], where a corientation of a (dynamic) graph G is used to answer adjacency queries in O(c) time using only linear space. These techniques [2,3,4] were further generalized to answer short-path queries =-=[5]-=-. Additional examples for the algorithmic use of low-degree orientations include load balancing [6], maximal matchings [7], counting subgraphs in sparse graphs [8], prize-collecting TSPs and Steiner T...

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...itional examples for the algorithmic use of low-degree orientations include load balancing [6], maximal matchings [7], counting subgraphs in sparse graphs [8], prize-collecting TSPs and Steiner Trees =-=[9]-=-, reporting all maximal independent sets [10], answering dominance queries [10], subgraph listing problems (listing triangles and 4-cliques) in planar graphs [2], and computing the girth [5]. Efficien...

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...ralized to answer short-path queries [5]. Additional examples for the algorithmic use of low-degree orientations include load balancing [6], maximal matchings [7], counting subgraphs in sparse graphs =-=[8]-=-, prize-collecting TSPs and Steiner Trees [9], reporting all maximal independent sets [10], answering dominance queries [10], subgraph listing problems (listing triangles and 4-cliques) in planar grap...

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...g” [1], where orientations are used to develop more efficient algorithms for finding simple cycles and paths. Another fundamental example is in data structures for quickly answering adjacency queries =-=[2,3,4]-=-, where a corientation of a (dynamic) graph G is used to answer adjacency queries in O(c) time using only linear space. These techniques [2,3,4] were further generalized to answer short-path queries [...

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...) time and deletions in worst-case O(1) time by using an O(log n)-orientation. These algorithms have been used as black-box components in several applications of dynamic graphs. Recently Gupta et.al. =-=[16]-=- showed that 4if only insertions are allowed then an amortized 2 edge reorientations suffice for maintaining a maximum out-degree of O(α(G)). Algorithms with amortized runtime bounds may be insuffici...

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...ned efficient amortized update time bounds, in contrast to our bounds which are all in the worst-case. Our results address an open question raised by Brodal and Fagerberg [3] and restated by Erickson =-=[15]-=-, of obtaining good worstcase bounds (although the ultimate goal is obviously worst-case time O(1) for all updates, if that is at all possible). 1.2 Comparison with Previous Work The dynamic setting i...