Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space TeX in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into TeX can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices

APPLICATION OF NEAR-INFRARED REFLECTANCE SPECTROSCOPY TO THE COMPOSITIONAL ANALYSIS OF BISCUITS AND BISCUIT DOUGHS

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space TeX in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into TeX can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices

Template-Driven Rainbow Coloring of Proper Interval Graphs

For efficient design of parallel algorithms on multiprocessor architectures with memory banks, simultaneous access to a specified subgraph of a graph data structure by multiple processors requires that the data items belonging to the subgraph reside in distinct memory banks. Such “conflict-free” access to parallel memory systems and other applied problems motivate the study of rainbow coloring of a graph, in which there is a fixed template T (or a family of templates), and one seeks to color the vertices of an input graph G with as few colors as possible, so that each copy of T in G is rainbow colored, i.e., has no two vertices the same color. In the above example, the data structure is modeled as the host graph G, and the specified subgraph as the template T. We call such coloring a template-driven rainbow coloring (or TR-coloring). For large data sets, it is also important to ensure that no memory bank (color) is overloaded, i.e., the coloring is as balanced as possible. Additionally, for fast access to data, it is desirable to quickly determine the address of a memory bank storing a data item. For arbitrary topology of G and T, finding an optimal and balanced TR-coloring is a challenging problem. This paper focuses on rainbow coloring of proper interval graphs (as hosts) for cycle templates. In particular, we present an O(k· | V| + | E| ) time algorithm to find a TR-coloring of a proper interval graph G with respect to k-length cycle template, Ck. Our algorithm produces a coloring that is (i) optimal, i.e., it uses minimum possible number of colors in any TR-coloring; (ii) balanced, i.e, the vertices are evenly distributed among the different color classes; and (iii) explicit, i.e., the color assigned to a vertex can be computed by a closed form formula in constant time

Improved algorithms for k-domination and total k-domination in proper interval graphs

Given a positive integer k, a k-dominating set in a graph G is a set of vertices such that every vertex not in the set has at least k neighbors in the set. A total k-dominating set, also known as a k-tuple total dominating set, is a set of vertices such that every vertex of the graph has at least k neighbors in the set. The problems of finding the minimum size of a k-dominating, resp. total k-dominating set, in a given graph, are referred to as k-domination, resp. total k-domination. These generalizations of the classical domination and total domination problems are known to be NP-hard in the class of chordal graphs, and, more specifically, even in the classes of split graphs (both problems) and undirected path graphs (in the case of total k-domination). On the other hand, it follows from recent work by Kang et al. (2017) that these two families of problems are solvable in time O(|V(G)|6k+4) in the class of interval graphs. In this work, we develop faster algorithms for k-domination and total k-domination in the class of proper interval graphs. The algorithms run in time O(|V(G)|3k) for each fixed k≥1 and are also applicable to the weighted case.Fil: Chiarelli, Nina. University of Primorska; EsloveniaFil: Hartinger, Tatiana Romina. University of Primorska; EsloveniaFil: Leoni, Valeria Alejandra. Universidad Nacional de Rosario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Lopez Pujato, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario; Argentina. Ministerio de Ciencia. Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica; ArgentinaFil: Milanič, Martin. University of Primorska; Esloveni