Bounds for identifying codes in terms of degree parameters

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If \M(G) denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then \M(G)\leq n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any $d\geq 3$, \M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$

Parameterized and approximation complexity of the detection pair problem in graphs

We study the complexity of the problem DETECTION PAIR. A detection pair of a graph $G$ is a pair $(W,L)$ of sets of detectors with $W\subseteq V(G)$, the watchers, and $L\subseteq V(G)$, the listeners, such that for every pair $u,v$ of vertices that are not dominated by a watcher of $W$, there is a listener of $L$ whose distances to $u$ and to $v$ are different. The goal is to minimize $|W|+|L|$. This problem generalizes the two classic problems DOMINATING SET and METRIC DIMENSION, that correspond to the restrictions $L=\emptyset$ and $W=\emptyset$, respectively. DETECTION PAIR was recently introduced by Finbow, Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The complexity of monitoring a network with both watchers and listeners. Manuscript, 2015], who proved it to be NP-complete on trees, a surprising result given that both DOMINATING SET and METRIC DIMENSION are known to be linear-time solvable on trees. It follows from an existing reduction by Hartung and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We show, using a reduction to SET COVER, that DETECTION PAIR is approximable within a factor logarithmic in the number of vertices of the input graph. Our two main results are a linear-time $2$-approximation algorithm and an FPT algorithm for DETECTION PAIR on trees.Comment: 13 page

Locating-total dominating sets in twin-free graphs: a conjecture

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $LT(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \geq 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $LT(G) \leq \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $LT(G) \le \frac{3}{4}n$.Comment: 18 pages, 1 figur

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

Random subgraphs make identification affordable

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph $G$ with $n$ vertices, maximum degree $\Delta=\omega(1)$ and minimum degree $\delta\geq c\log{\Delta}$, for some constant $c>0$, contains a large spanning subgraph which admits an identifying code with size $O\left(\frac{n\log{\Delta}}{\delta}\right)$. In particular, if $\delta=\Theta(n)$, then $G$ has a dense spanning subgraph with identifying code $O\left(\log n\right)$, namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code

Locating-dominating sets and identifying codes in graphs of girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.Comment: 20 pages, 9 figure

Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph $G$, that is, a set $L$ of vertices such that all vertices in $G$ are uniquely determined by their relative distances to the vertices of $L$. A centroidal locating set of $G$ of minimum size is called a centroidal basis, and its size is the centroidal dimension $CD(G)$. This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric dimension and twice the location-domination number of $G$, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph $G$ with $n$ vertices and maximum degree at least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, $CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and cycles. We finally investigate the computational complexity of determining $CD(G)$ for an input graph $G$, showing that the problem is hard and cannot even be approximated efficiently up to a factor of $o(\log n)$. We also give an $O\left(\sqrt{n\ln n}\right)$-approximation algorithm

On powers of interval graphs and their orders

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl

Bounds and extremal graphs for total dominating identifying codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached). Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains this bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the related parameter $\gamma^{\text{ID}}(G)$ as well as the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight