First-Fit is Linear on Posets Excluding Two Long Incomparable Chains

A poset is (r + s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo

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

On Assigning Prefix Free Codes to the Vertices of a Graph

For a graph G on n vertices, with positive integer weights $w_1, . . . , w_n$ assigned to the n vertices such that, for every clique K of G, \[ \sum_{i \in K}{\frac{1}{2^{w_i}}} \leq 1 \], the problem we are interested in is to assign binary codes $C_1, . . . , C_n$ to the vertices such that $C_i$ has $w_i$ (or a function of $w_i$) bits in it and, for every edge \{i, j\}, $C_i$ and $C_j$ are not prefixes of each other.We call this the Graph Prefix Free Code Assignment Problem. We relate this new problem to the problem of designing adversaries for comparison based sorting algorithms. We show that the decision version of this problem is as hard as graph colouring and then present results on the existence of these codes for prefect graphs and its subclasses

On the Structure of Contractible Edges in k-connected Partial k-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k - 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees

On the Structure of Contractible Edges in k-connected Partial k-trees

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k - 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees

Solving MIN ONES 2-SAT as fast as VERTEX COVER

The problem of finding a satisfying assignment that minimizes the number of variables that are set to 1 is NP-complete even for a satisfiable 2-SAT formula. We call this problem MIN ONES 2-SAT. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k. In this paper, we present a polynomial-time reduction from MIN ONES 2-SAT to VERTEX COVER without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k - c logk-variable kernel subsuming (or, in the case of kernels, improving) the results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover. Finally we show that the optimum value of the LP relaxation of the MIN ONES 2-SAT and that of the corresponding VERTEX COVER are the same. This implies that the (recent) results of VERTEX COVER version parameterized above the optimum value of the LP relaxation of VERTEX COVER carry over to the MIN ONES 2-SAT version parameterized above the optimum of the LP relaxation of MIN ONES 2-SAT. (C) 2013 Elsevier B.V. All rights reserved

Aggregation and photoresponsive behavior of azobenzene–oligomethylene–glucopyranoside bolaamphiphiles

The self-aggregation behavior of a series of photoresponsive bolaamphiphiles consisting of a central azobenzene moiety linked to terminal sugar groups via oligomethylene spacers in water/DMSO mixed solvents are reported. Changes in the absorption spectra in these solvent mixtures indicated the formation of tightly packed H-aggregates. The ease of formation and the stability of the aggregates were observed to increase with increase in the length of the oligomethylene spacers. Microscopic studies indicated the formation of vesicles or fibers depending upon the length of the oligomethylene spacers. Disruption of the aggregates induced by photoisomerization of the azobenzene unit and their reformation via the subsequent thermal cis to trans isomerization was also investigated

Max-coloring of vertex-weighted graphs

[[abstract]]A proper vertex coloring of a graph G is a partition \{A_1,A_2,\ldots ,A_k\} of the vertex set V(G) into stable sets. For a graph G with a positive vertex-weight c:V(G) \rightarrow (0,\infty ), denoted by (G,c), let \chi (G,c) be the minimum value of \sum _{i=1}^k \max _{v \in A_i} c(v) over all proper vertex coloring \{A_1,A_2,\ldots ,A_k\} of G and \sharp \chi (G,c) the minimum value of k for a proper vertex coloring \{A_1,A_2,\ldots ,A_k\} of G such that \sum _{i=1}^k \max _{v \in A_i} c(v) = \chi (G,c). This paper establishes an upper bound on \sharp \chi (G,c) for a weighted r-colorable graph (G,c), and a Nordhaus–Gaddum type inequality for \chi (G,c). It also studies the c-perfection for a weighted graph (G,c).[[notice]]補正完