The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either V(G) \ C or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

In this paper, we introduce and study a new scoring game on graphs called smash and grab. In this game, two players, called Left and Right, take turns removing a vertex of the graph as well as all of its neighbours that become isolated by this removal. For each player and each of their turns, they score the number of vertices that were removed on their turn. The game ends when there are no more vertices remaining, and the player with the highest final score wins. We denote by Ls(G) the difference between Left and Right's final scores in G when Left starts and both players play optimally (they both aim to maximise their scores). We mainly study this parameter for different graph classes. We notably prove that Ls(F)≥0 for any forest F (i.e., the first player cannot lose). We then use this result to compute the exact value of Ls(G) for particular forests such as unions of paths and subdivided stars. The result in paths then solves the case of a unique cycle. Finally, we prove that, for a generalisation of the game, computing the score is PSPACE-complete.

A dominating set D in a digraph is a set of vertices such that every vertex is either in D or has an in-neighbour in D. A dominating set D of a digraph is locating-dominating if every vertex not in D has a unique set of in-neighbours within D. The location-domination number γL(G) of a digraph G is the smallest size of a locating-dominating set of G. We investigate upper bounds on γL(G) in terms of the order of G. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with the same (open or closed) in-neighbourhoods. Thus, we investigate the value of γL(G) in the absence of twins and give a general method for constructing small locating-dominating sets by the means of special dominating sets. In this way, we show that for every twin-free digraph G of order n, γL(G)≤4n5+1 holds, and there exist twin-free digraphs G with γL(G)=2(n−2)3. Improved bounds are proved for certain special cases. In particular, if G is twin-free and a tournament, or twin-free and acyclic, we prove γL(G)≤⌈n2⌉, which is tight in both cases.

The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter $d$ and metric dimension $k$. In general, the bound $n\leq d^k+k$ is known to hold. We prove a bound of the form $n=\mathcal{O}(kd^2)$ for trees and outerplanar graphs (for trees we determine the best possible bound and the corresponding extremal examples). More generally, for graphs having a tree decomposition of width $w$ and length $\ell$, we obtain a bound of the form $n=\mathcal{O}(kd^2(2\ell+1)^{3w+1})$. This implies in particular that $n=\mathcal{O}(kd^{\mathcal{O}(1)})$ for graphs of constant treewidth and $n=\mathcal{O}(f(k)d^2)$ for chordal graphs, where $f$ is a doubly exponential function. Using the notion of distance-VC dimension (introduced in 2014 by Bousquet and Thomassé) as a tool, we prove the bounds $n\leq (dk+1)^{t-1}+1$ for $K_t$-minor-free graphs and $n\leq (dk+1)^{d(3\cdot 2^{r}+2)}+1$ for graphs of rankwidth at most $r$.

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class $\mathcal{C}$, that is, the maximum VC-dimension over the hypergraphs formed by the closed neighborhoods of elements of $\mathcal{C}$. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in the number of vertices, while classes with finite VC-dimension have a polynomial lower bound. We then turn to approximation algorithms. We show that Min Id Code (the problem of finding a smallest identifying code in a given graph from some class $\mathcal{C}$) is log-APX-hard for any hereditary class of infinite VC-dimension. For hereditary classes of finite VC-dimension, the only known previous results show that we can approximate Min Id Code within a constant factor in some particular classes, e.g., line graphs, planar graphs, and unit interval graphs. We prove that Min Id Code can be approximate within a factor 6 for interval graphs. In contrast, we show that Min Id Code on $C_4$-free bipartite graphs (a class of finite VC-dimension) cannot be approximated to within a factor of $c \log(|V|)$ for some $c>0$.

Abstract The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. The identifying codes can be applied, for example, to sensor networks. In this article, we consider as sensors the set \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document} $\mathbb{Z}^2$ \end{document} where one sensor can check its neighbors within Euclidean distance r. We construct tolerant identifying codes in this network that are robust against some changes in the neighborhood monitored by each sensor. We give bounds for the smallest density of a tolerant identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

Karpovsky, Chakrabarty and Levitin introduced identifying codes, which can be applied, for example, to locating objects in sensor networks. In this paper, the underlying structure is Z2where one sensor can check its neighbours within Euclidean distance r. We construct identifying codes in this network that are robust against some changes in the neighbourhood monitored by each sensor. We give bounds for the smallest density of such an identifying code for general values of r. We also provide infinite families of values r with optimal such codes and study the case of small values of r.