Let = (,) be a connected graph with at least two vertices. These concepts were studied by Chartrand. The detour number () is called a minimum order of a detour set and any detour set of order () is called a minimum detour set of. A set ⊆ is called a detour set if every vertex in lies on a detour joining a pair of vertices of. A vertex is said to lie on a − detour if is a vertex of a − detour path including the verticles and. It is known that the detour distance is a metric on the vertex set (). A − path of length (,) is called a − detour. For vertices and in a connected graph, the detour distance (,) is the length of the longest − path in G. For basic definitions and terminologies, we refer to. We consider connected graphs with atleast two vertices. The order and size of are denoted by and respectively. For any graph, the set of vertices is denoted by () and the edge set by (). We consider finite graphs without loops and multiple edges. and derive the same of some standard and special graphs. We also introduce the new concept efficiently dominationating detour number, characterize it. In this paper, we study the detour domination number of some special graphs. A dominating set of is a subset of () such that every vertex in − is adjacent to some vertex in. ![]() A subset of is called a detour set of G if every vertex in − lies in a detour joining a pair of vertices of. In particular, we define a notion of k-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph G is 2Δ(G)-matroidally colorable. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on "independent systems of representatives" (which are the special case in which the matroid is a partition matroid). Bounds are also found on the dual parameter - the maximal number of disjoint sets, all spanning in each of two given matroids. This, in turn, is used to derive a weakened version of a conjecture of Rota. belonging to the intersection of two matroids, needed to cover their common ground set. Another is an upper bound on the minimal number of sets. One application is a solution of the case r = 3 of a matroidal version of Ryser's conjecture. We generalize this theorem, replacing one of the matroids by a general simplicial complex. This makes the new algorithm one of the most efficient solutions in practical cases.Ī classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a "covering" parameter. The gain in speed-up, in terms of running times, is up to 48%. According to experimental results, our algorithm obtains in most practical cases the best running times, when compared against the most effective solutions. Although our solution has a quadratic worst-case time complexity, it shows a sub-linear behaviour on average. In this paper, we present a new efficient algorithm for the swap matching problem based on character comparison and structured as a generalization of the Skip-Search algorithm for the exact string matching problem. In recent years, much research has focused on practical solutions and efficient algorithms have been devised by means of the bit-parallel simulation of non-deterministic automata. solved the problem in O(n log m log σ) worst-case time complexity, where σ is the size of the alphabet. The swap matching problem consists in finding all occurrences of a pattern x of length m in a text y of length n, allowing for disjoint local swaps of characters in the pattern. ![]() The index of f is bounded in terms of its length and the bounds are shown tight by examples. The minimal length of two words which witness the non-isometricity of a word f is called its index. Then, Ham-isometric and Lee-isometric k-ary words are characterized in terms of their overlaps with errors. From k≥4, the isometricity in terms of cubes is no longer captured by the Ham-isometricity, but by the Lee-isometricity. The case of a k-ary alphabet, with k≥2, is here investigated. A binary word f is isometric if and only if it is Ham-isometric, i.e., for any pair of f-free binary words u and v, u can be transformed in v by complementing the bits on which they differ and generating only f-free words. When such a subgraph is isometric to the cube, for any n≥1, the word f is said isometric. Given a k-ary word f, the k-ary n-cube avoiding f is the subgraph obtained deleting those vertices which contain f as a factor. More precisely, a k-ary n-cube is a graph with kn vertices associated to the k-ary words of length n. The k-ary n-cubes are a generalization of the hypercubes to alphabets of cardinality k, with k≥2.
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