Criteria for interpreting kimberlite as coherent: Insights from the Muskox and Jericho kimberlites (Nunavut, Canada)

The Jurassic Muskox and Jericho kimberlites (Northern Slave Province, Nunavut, Canada) contain a variety of facies exhibiting different geometries, contact relationships, internal organisation, country rock abundance and olivine shapes, although many have similar matrix/groundmass mineralogies and textures. Five facies are examined that either have characteristics consistent with coherent rocks in general (i.e. intrusive and extrusive non-fragmental rocks) or are mineralogically and texturally similar to kimberlite described as coherent (or apparent coherent). Three facies are interpreted as coherent on the basis of: (1) geological setting, (2) apparent-porphyritic texture, (3) sharp contacts with fragmental kimberlite, (4) relative abundance of elongate and unbroken olivine crystals and (5) paucity of country rock xenoliths, while the remaining two facies are interpreted as fragmental on the basis of: (1) the gradational contacts with demonstrably fragmental kimberlite, (2) relative abundance and range of sizes of country rock lithic clasts and (3) numerous broken olivine crystals. Comparisons are made with coherent and apparent-coherent kimberlite from the literature. Our three coherent facies are similar to literature reported coherent kimberlite dykes hosted in country rock (CKd) in terms of internal organisation, low abundance of country rock xenoliths, and apparent-porphyritic texture. Conversely, our two fragmental facies share attributes with previously described pipe-filling coherent and apparent-coherent kimberlite (CKpf) in terms of geometry, internal organisation and abundance of country rock xenoliths. We conclude that CKd and most CKpf, although similar in matrix/groundmass mineralogy and texture, can be distinguished on the basis of internal organisation, country rock lithic clast abundance, texture (e.g. apparent-porphyritic texture) and possibly olivine crystal shapes and suggest that fragmental kimberlite is more common than reported.

Efficient Output-Sensitive Construction of Reeb Graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. This paper describes a near-optimal two-step algorithm that constructs the Reeb graph of a Morse function defined over manifolds in any dimension. The algorithm first identifies the critical points of the input manifold, and then connects these critical points in the second step to obtain the Reeb graph. A simplification mechanism based on topological persistence aids in the removal of noise and unimportant features. A radial layout scheme results in a feature-directed drawing of the Reeb graph. Experimental results demonstrate the efficiency of the Reeb graph construction in practice and its applications.

On the cubicity of bipartite graphs

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k, where each R-i is a closed interval on the real line of the form [a(j), a(i), + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for graph G, cub(G) <= left perpendicular2n/3right perpendicular. Recently it has been shown that for a graph G, cub(G) >= 4(Delta + 1) In n, where n and Delta are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G = (A boolean OR B, E) with |A| = n(1), |B| = n2, n(1) <= n(2), and Delta' = min {Delta(A),Delta(B)}, where Delta(A) = max(a is an element of A)d(a) and Delta(B) = max(b is an element of B) d(b), d(a) and d(b) being the degree of a and b in G, respectively , cub(G) <= 2(Delta' + 2) bar left rightln n(2)bar left arrow. We also give an efficient randomized algorithm to construct the cube representation of G in 3 (Delta' + 2) bar right arrowIn n(2)bar left arrow dimension. The reader may note that in general Delta' can be much smaller than Delta.

Cubicity of threshold graphs

We show that the cubicity of a connected threshold graph is equal to inverted right perpendicularlog(2) alpha inverted left perpendicular, where alpha is its independence number.

Characterisation of k-Variegated Graphs, k greater-than-or-equal-to 3

A graph is said to be k-variegated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it. Bednarek and Sanders [1] posed the problem of characterizing k-variegated graphs. V.N. Bhat-Nayak, S.A. Choudum and R.N. Naik [2] gave the characterization of 2-variegated graphs. In this paper we characterize k-variegated graphs for k greater-or-equal, slanted 3.

A Touch of Red : Archaeological and Ethnographic Approaches to Interpreting Finnish Rock Paintings

Approximately 125 prehistoric rock paintings have been found in the modern territory of Finland. The paintings were done with red ochre and are almost without exception located on steep lakeshore cliffs associated with ancient water routes. Most of the sites are found in the central and eastern parts of the country, especially on the shores of Lakes Päijänne and Saimaa. Using shore displacement chronology, the art has been dated to ca. 5000 – 1500 BC. It was thus created mainly during the Stone Age and can be associated with the so-called ‘Comb Ware’ cultures of the Subneolithic period. The range of motifs is rather limited, consisting mainly of schematic depictions of stick-figure humans, elks, boats, handprints and geometric signs. Few paintings include any evidence of narrative scenes, making their interpretation a rather difficult task. In Finnish archaeological literature, the paintings have traditionally been associated with ’sympathetic’ hunting magic, or the belief that the ritual shooting of the painted animals would increase hunting luck. Some writers have also suggested totemistic and shamanistic readings of the art. This dissertation is a critical review of the interpretations offered of Finnish rock art and an exploration of the potentials of archaeological and ethnographic research in increasing our knowledge of its meaning. Methods used include ’formal’ approaches such as archaeological excavation, landscape analysis and the application of neuropsychological research to the study of rock art, as well as ethnographically ’informed’ approaches that make use of Saami and Baltic Finnish ethnohistorical sources in interpretation. In conclusion, it is argued that although North European hunter-gatherer rock art is often thought to lie beyond the reach of ‘informed’ knowledge, the exceptional continuity of prehistoric settlement in Finland validates the informed approach in the interpretation of Finnish rock paintings. The art can be confidently associated with shamanism of the kind still practiced by the Saami of Northern Fennoscandia in the historical period. Evidence of similar shamanistic practices, concepts and cosmology are also found in traditional Finnish-Karelian epic poetry. Previous readings of the art based on ‘hunting magic’ and totemism are rejected. Most of the paintings appear to depict experiences of falling into a trance, of shamanic metamorphosis and trance journeys, and of ‘spirit helper’ beings comparable to those employed by the Saami shaman (noaidi). As demonstrated by the results of an excavation at the rock painting of Valkeisaari, the painted cliffs themselves find a close parallel in the Saami cult of the 'sieidi', or sacred cliffs and boulders worshipped as expressing a supernatural power. Like the Saami, the prehistoric inhabitants of the Finnish Lake Region seem to have believed that certain cliffs were ’alive’ and inhabited by the spirit helpers of the shaman. The rock paintings can thus be associated with shamanic vision quests, and the making of ‘art’ with an effort to socialize the other members of the community, especially the ritual specialists, with trance visions. However, the paintings were not merely to be looked at. The red ochre handprints pressed on images of elks, as well as the fact that many paintings appear ’smeared’, indicate that they were also to be touched – perhaps in order to tap into the supernatural potency inherent in the cliff and in the paintings of spirit animals.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

On the Cubicity of Interval Graphs

A k-cube (or ``a unit cube in k dimensions'') is defined as the Cartesian product R-1 x . . . x R-k where R-i (for 1 <= i <= k) is an interval of the form [a(i), a(i) + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i. e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Delta, cub(G) <= inverted right perpendicular log(2) Delta inverted left perpendicular + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to inverted right perpendicular log(2) Delta inverted left perpendicular.

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K-4, then box(G) = 2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G) = 2 unless G is isomorphic to K4 (in which case its boxicity is 1).

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

Acyclic Edge Coloring of Graphs with Maximum Degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a'(G) <= Delta+2, where Delta=Delta(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Delta(G)<= 4, with the additional restriction that m <= 2n-1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m <= 2n, when Delta(G)<= 4. It follows that for any graph G if Delta(G)<= 4, then a'(G) <= 7.

Representation of functional dependencies in relational databases using linear graphs

Functional dependencies in relational databases are investigated. Eight binary relations, viz., (1) dependency relation, (2) equipotence relation, (3) dissidence relation, (4) completion relation, and dual relations of each of them are described. Any one of these eight relations can be used to represent the functional dependencies in a database. Results from linear graph theory are found helpful in obtaining these representations. The dependency relation directly gives the functional dependencies. The equipotence relation specifies the dependencies in terms of attribute sets which functionally determine each other. The dissidence relation specifies the dependencies in terms of saturated sets in a very indirect way. Completion relation represents the functional dependencies as a function, the range of which turns out to be a lattice. Depletion relation which is the dual of the completion relation can also represent functional dependencies and similarly can the duals of dependency, equipotence, and dissidence relations. The class of depleted sets, which is the dual of saturated sets, is defined and used in the study of depletion relations.

An (O)over-tilde(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs

We present a fast algorithm for computing a Gomory-Hu tree or cut tree for an unweighted undirected graph G = (V, E). The expected running time of our algorithm is (O) over tilde (mc) where vertical bar E vertical bar = m and c is the maximum u-v edge connectivity, where u, v is an element of V. When the input graph is also simple (i.e., it has no parallel edges), then the u-v edge connectivity for each pair of vertices u and v is at most n - 1; so the expected run-ning time of our algorithm for simple unweighted graphs is (O) over tilde (mn). All the algorithms currently known for constructing a Gomory-Hu tree [8, 9] use n - 1 minimum s-t cut (i.e., max flow) subroutines. This in conjunction with the current fastest (O) over tilde (n(20/9)) max flow algorithm due to Karger and Levine[11] yields the current best running time of (O) over tilde (n(20/9)n) for Gomory-Hu tree construction on simple unweighted graphs with m edges and n vertices. Thus we present the first (O) over tilde (mn) algorithm for constructing a Gomory-Hu tree for simple unweighted graphs. We do not use a max flow subroutine here; we present an efficient tree packing algorithm for computing Steiner edge connectivity and use this algorithm as our main subroutine. The advantage in using a tree packing algorithm for constructing a Gomory-Hu tree is that the work done in computing a minimum Steiner cut for a Steiner set S subset of V can be reused for computing a minimum Steiner cut for certain Steiner sets S' subset of S.

On the Arrangement of Cliques in Chordal Graphs with respect to the Cuts

A cut (A, B) (where B = V - A) in a graph G = (V, E) is called internal if and only if there exists a vertex x in A that is not adjacent to any vertex in B and there exists a vertex y is an element of B such that it is not adjacent to any vertex in A. In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut (A, B) in a chordal graph G, there exists a clique with kappa(G) + vertices (where kappa(G) is the vertex connectivity of G) such that it is (approximately) bisected by the cut (A, B). In fact we give a stronger result: For any internal cut (A, B) of a chordal graph, and for each i, 0 <= i <= kappa(G) + 1 such that vertical bar K-i vertical bar = kappa(G) + 1, vertical bar A boolean AND K-i vertical bar = i and vertical bar B boolean AND K-i vertical bar = kappa(G) + 1 - i. An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be Omega(k(2)), where kappa(G) = k. Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least kappa(G)(kappa(G)+1)/2 where kappa(G) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to kappa(G). This result is tight.