Y. L. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8(1):99--118, 1995. 1  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is that it does not exploit symmetry. Later Coleman and More [2] addressed this issue and showed that the resulting problem (SYMCOLPART) is equivalent to the D coloring problem. 4 3 Parallel Coloring Algorithms The distance k coloring problem is NP hard for any fixed integer k [12]. A proof sketch showing that D coloring is NP hard is given in [2] Furthermore, Lexicographically First # 1 Coloring (LFC) the polynomial variant of D1 coloring in which the vertices are given in a predetermined order and the question at each step is to assign the vertex the smallest color ....

Y. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM J. Disc. Math., 8:99--118, 1995.

.... Gamma f(y)j 2d and whenever the distance between x and y is two, jf(x) Gamma f(y)j d. It is shown that it suffices to consider integral valued labelings. We denote (G; 1) L(G; 2; 1) by (G) The version of the problem where the labels that are used are in a consecutive range, was studied in [18]. As far as special graph classes are concerned, G) can be determined efficiently for paths, cycles and wheels [9] Bounds for cubes were obtained in [11, 9, 19] It is conjectured that (Qn ) n 3 for n 3 in [9] Non trivial algorithms were also found for cographs [4] In [11] planar graphs ....

....: d k ) problem. For the sake of history (and also for convenience) we write (G) instead of L(G; 2; 1) Definition 2. A labeling is called consecutive if the labels that are used are consecutive. Clearly, a consecutive labeling does not always exist. Consecutive labelings were studied in [18]. In general, it can be shown [18] that G = V; E) has a Hamiltonian path if and only if G (the complement of G) has a consecutive (2;1) labeling with jV j Gamma 1. In this paper we show that the fixed parameter variant with 4 is equally difficult. Most of our NP hardness results are ....

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Lin, Yaw-Ling and Steven S. Skiena, Algorithms for square roots of graphs, SIAM J. Discrete Math. 8, (1995), pp. 99--118.

....trees, interval graphs, and strongly chordal graphs have received special attention. We note that every strongly chordal graph and every split graph is chordal. Also, every tree and every interval graph is strongly chordal. The fact that any power of a tree is chordal and proof thereof appears in [16] and [4] However, Robert Jamison (Personal Communication, 2000) at Clemson University may have been the first to prove that property in the early eighties. Linear time algorithms are given in [16] for finding a tree square root of a given graph and a square root of a planar graph. In [4] a ....

....graph is strongly chordal. The fact that any power of a tree is chordal and proof thereof appears in [16] and [4] However, Robert Jamison (Personal Communication, 2000) at Clemson University may have been the first to prove that property in the early eighties. Linear time algorithms are given in [16] for finding a tree square root of a given graph and a square root of a planar graph. In [4] a polynomial time algorithm for recognizing tree powers is given as well as a short proof that any power of a tree is strongly chordal. In [19] and [5] it is shown that any power of a strongly chordal ....

[Article contains additional citation context not shown here]

Y.-L. Lin and S. Skiena. Algorithms for Square Roots of Graphs. SIAM Journal of Discrete Mathematics, 8(1):99--118, (1995).

....this to be an upper bound on the chromatic number, for large. Some work has been done on the case = 3, as listed in [5, Problem 2. 18] McCormick [9] showed that the problem of coloring the power of a graph is NP complete, for any xed power, and a later proof was given by Lin and Skiena [8]. McCormick gave a greedy algorithm with a O( p n) approximation for squares of general graphs. Heggernes and Telle [4] showed that determining if the square of a cubic graph can be colored with 4 colors or less is NP complete, while it is easily determined if 3 colors suce. Ramanathan and Lloyd ....

Y.-L. Lin and S. Skiena. Algorithms for square roots of graphs. SIAM J. Disc. Math., 1995.

....trees, interval graphs, and strongly chordal graphs have received special attention. We note that every strongly chordal graph and every split graph is chordal. Also, every tree and every interval graph is strongly chordal. The fact that any power of a tree is chordal and proof thereof appears in [16] and [4] However, Robert Jamison (Personal Communication, 2000) at Clemson University may have been the rst to prove that property in the early eighties. Linear time algorithms are given in [16] for nding a tree square root of a given graph and a square root of a planar graph. In [4] a ....

....graph is strongly chordal. The fact that any power of a tree is chordal and proof thereof appears in [16] and [4] However, Robert Jamison (Personal Communication, 2000) at Clemson University may have been the rst to prove that property in the early eighties. Linear time algorithms are given in [16] for nding a tree square root of a given graph and a square root of a planar graph. In [4] a polynomial time algorithm for recognizing tree powers is given as well as a short proof that any power of a tree is strongly chordal. In [19] and [5] it is shown that any power of a strongly chordal ....

[Article contains additional citation context not shown here]

Y.-L. Lin and S. Skiena. Algorithms for Square Roots of Graphs. SIAM Journal of Discrete Mathematics, 8(1):99-118, (1995).

....to be an upper bound on the chromatic number, for Delta large. Some work has been done on the case Delta = 3, as listed in [5, Problem 2. 18] McCormick [9] showed that the problem of coloring the power of a graph is NP complete, for any fixed power, and a later proof was given by Lin and Skiena [8]. McCormick gave a greedy algorithm that gives a O( p n) approximation for squares of general graphs. Heggernes and Telle [4] showed that determining if the square of a cubic graph can be colored with 4 colors or less is NP complete, while it is easily determined if 3 colors suffice. Ramanathan ....

....and a set of N vertices attached to the end node of each path. For the case of d = 2t, we use a construction similar to the theorem above, except a path of t Gamma 1 vertices lies between each v i vertex and the corresponding u i;x vertices. We note that NP hardness reduction of Lin and Skiena [8] yields nearly the same result, or a (n=d) 1=2 Gammaffl hardness. On the positive side, we can obtain nontrivial approximation for coloring all power graphs. In contrast with the Independent Set problem [3] the coloring problem becomes easier in odd powers. Theorem 4.4 Coloring G 2t Gamma1 ....

Y.-L. Lin and S. Skiena. Algorithms for square roots of graphs. SIAM J. Disc. Math., 1995.

....# in G. Our approximation ratio cannot be polynomially improved in the sense that no (n 4) 1 2 # approximation, for any fixed # 0, can be obtained in polynomial time unless NP = ZPP. Studying NP hard problems in powers of graphs is a topic that has received some attention in the literature [5, 34, 18, 4]. Independently of our work, Baveja and Srinivasan (personal communication) have obtained results similar to ours for approximating vertex disjoint paths under the rounding approach, unsplittable flow and column restricted packing integer programs. Their work builds on the methods in [31] 2 ....

Y.-L. Lin and S. E. Skiena. Algorithms for square roots of graphs. SIAM J. on Discrete Mathematics, 8(1):99--118, February 1995.

....the case where the power of a graph belongs to a special class. Ross Harary [10] showed that the tree square roots of a graph, when they exist, are unique up to isomorphisms. Harary, Karp Tutte [5] provided characterizations of the planar graphs which are square graphs. Recently, Lin Skiena [7] devised several algorithms with respect to powers of graphs. They provided efficient algorithms for finding square roots of graphs G 2 where G is a tree, and where G 2 is planar. They also presented several polynomial time algorithms for problems which are NP complete in general, when ....

Y-L. Lin and S.S. Skiena, "Algorithms for Square Roots of Graphs," Technical Report TR# 91/11, Department of Computer Science, SUNY Stony Brook, 1991.

.... 3 It is NP complete to recognize a graph power [5] but it is possible to determine if a graph has a k root tree, for any fixed k, in O(n 3 ) time, where n is the number of vertices in the input graph [3] For the special case k = 2, determining if a graph has a 2 root tree takes O(n e) time [4], where e is the number of edges in the input graph. There is rich literature on graph roots and powers (see [1] for an overview) but few results on phylogenetic roots powers and Steiner roots powers. Recently, Nishimura, Ragde, and Thilikos [6] presented an O(n 3 ) time algorithm for a variant ....

Y.-L. Lin and S.S. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8:99--118, 1995.

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Y. L. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8(1):99--118, 1995. 1

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Y.-L. Lin and S. E. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8(1):99-118, February 1995.

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Y.-L. Lin and S. E. Skiena. Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics, 8(1):99--118, February 1995.

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Y. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM J. Disc. Math., 8:99--118, 1995.

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