Mod/Resc Parsimony Inference

We address in this paper a new computational biology problem that aims at understanding a mechanism that could potentially be used to genetically manipulate natural insect populations infected by inherited, intra-cellular parasitic bacteria. In this problem, that we denote by \textsc{Mod/Resc Parsimony Inference}, we are given a boolean matrix and the goal is to find two other boolean matrices with a minimum number of columns such that an appropriately defined operation on these matrices gives back the input. We show that this is formally equivalent to the \textsc{Bipartite Biclique Edge Cover} problem and derive some complexity results for our problem using this equivalence. We provide a new, fixed-parameter tractability approach for solving both that slightly improves upon a previously published algorithm for the \textsc{Bipartite Biclique Edge Cover}. Finally, we present experimental results where we applied some of our techniques to a real-life data set.Comment: 11 pages, 3 figure

Discussion On Some Important Topics in Graph Theory

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Some Investigations in the Theory of Graphs

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Some Topics of Special Interest in Graph Theory

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Games on graphs, visibility representations, and graph colorings

In this thesis we study combinatorial games on graphs and some graph parameters whose consideration was inspired by an interest in the symmetry of hypercubes. A capacity function f on a graph G assigns a nonnegative integer to each vertex of V(G). An f-matching in G is a set M ⊆ E(G) such that the number of edges of M incident to v is at most f(v) for all v ⊆ V(G). In the f-matching game on a graph G, denoted (G,f), players Max and Min alternately choose edges of G to build an f-matching; the game ends when the chosen edges form a maximal f-matching. Max wants the final f-matching to be large; Min wants it to be small. The f-matching number is the size of the final f-matching under optimal play. We extend to the f-matching game a lower bound due to Cranston et al. on the game matching number. We also consider a directed version of the f-matching game on a graph G. Peg Solitaire is a game on connected graphs introduced by Beeler and Hoilman. In the game, pegs are placed on all but one vertex. If x, y, and z form a 3-vertex path and x and y each have a peg but z does not, then we can remove the pegs at x and y and place a peg at z; this is called a jump. The goal of the Peg Solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph G when no jumps remain is the Fool's Solitaire number F(G). We determine the Fool's Solitaire number for the join of any graphs G and H. For the cartesian product, we determine F(G ◻ K_k) when k ≥ 3 and G is connected. Finally, we give conditions on graphs G and H that imply F(G ◻ H) ≥ F(G) F(H). A t-bar visibility representation of a graph G assigns each vertex a set that is the union of at most t horizontal segments ("bars") in the plane so that vertices are adjacent if and only if there is an unobstructed vertical line of sight (having positive width) joining the sets assigned to them. The visibility number of a graph G, written b(G), is the least t such that G has a t-bar visibility representation. Let Q_n denote the n-dimensional hypercube. A simple application of Euler's Formula yields b(Q_n) ≥ ⌈(n+1)/4⌉. To prove that equality holds, we decompose Q_{4k-1} explicitly into k spanning subgraphs whose components have the form C_4 ◻ P_{2^l}. The visibility number b(D) of a digraph D is the least t such that D can be represented by assigning each vertex at most t horizontal bars in the plane so that uv ∈ E(D) if and only if there is an unobstructed vertical line of sight (with positive width) joining some bar for u to some higher bar for v. It is known that b(D) ≤ 2 for every outerplanar digraph. We give a characterization of outerplanar digraphs with b(D)=1. A proper vertex coloring of a graph G is r-dynamic if for each v ∈ V (G), at least min{r, d(v)} colors appear in N_G(v). We investigate r-dynamic versions of coloring and list coloring. We give upper bounds on the minimum number of colors needed for any r in terms of the genus of the graph. Two vertices of Q_n are antipodal if they differ in every coordinate. Two edges uv and xy are antipodal if u is antipodal to x and v is antipodal to y. An antipodal edge-coloring of Q_n is a 2-coloring of the edges in which antipodal edges have different colors. DeVos and Norine conjectured that for n ≥ 2, in every antipodal edge-coloring of Q_n there is a pair of antipodal vertices connected by a monochromatic path. Previously this was shown for n ≤ 5. Here we extend this result to n = 6. Hovey introduced A-cordial labelings as a simultaneous generalization of cordial and harmonious labelings. If S is an abelian group, then a labeling f: V(G) → A of the vertices of a graph G induces an edge-labeling on G; the edge uv receives the label f(u) + f(v). A graph G isA-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most 1, and (2) the induced edge label classes differ in size by at most 1. The smallest non-cyclic group is V_4 (also known as Z_2×Z_2). We investigate V_4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q

Betweenness centrality in convex amalgamation of graphs

Betweenness centrality measures the potential or power of a node to control the communication over the network under the assumption that information flows primarily over the shortest paths between pair of nodes. The removal of a node with highest betweenness from the network will most disrupt communications between other nodes because it lies on the largest number of paths. A large network can be thought of as inter-connection between smaller networks by means of different graph operations. Thus the structure of a composite graph can be studied by analysing its component graphs. In this paper we present the betweenness centrality of some classes of composite graphs constructed by the graph operation called amalgamation or merging

Conflict in a peacekeeping organization: UNFICYP

[Δε διατίθεται περίληψη][No abstract available