Location-domination in line graphs

A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$, where $N(u)$ denotes the open neighborhood of $u$. If $D$ is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of $G$. A graph $G$ is twin-free if every two distinct vertices of $G$ have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph $G$ without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure

Hybrid tractability of soft constraint problems

The constraint satisfaction problem (CSP) is a central generic problem in computer science and artificial intelligence: it provides a common framework for many theoretical problems as well as for many real-life applications. Soft constraint problems are a generalisation of the CSP which allow the user to model optimisation problems. Considerable effort has been made in identifying properties which ensure tractability in such problems. In this work, we initiate the study of hybrid tractability of soft constraint problems; that is, properties which guarantee tractability of the given soft constraint problem, but which do not depend only on the underlying structure of the instance (such as being tree-structured) or only on the types of soft constraints in the instance (such as submodularity). We present several novel hybrid classes of soft constraint problems, which include a machine scheduling problem, constraint problems of arbitrary arities with no overlapping nogoods, and the SoftAllDiff constraint with arbitrary unary soft constraints. An important tool in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page