Locally identifying coloring in bounded expansion classes of graphs

A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.]

The switch operators and push-the-button games: a sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para{\^i}tr

A new approach to the 22-regularity of the ℓ\ell-abelian complexity of 22-automatic sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.Comment: 44 pages, 2 figures; publication versio

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

On three domination numbers in block graphs

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view. Hence, a typical line of attack for these problems is to determine lower and upper bounds for minimum codes in special graphs. In this work we study the problem of determining the cardinality of minimum codes in block graphs (that are diamond-free chordal graphs). We present for all three codes lower and upper bounds as well as block graphs where these bounds are attained

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Domination and location in twin-free digraphs

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 $\gamma_L(G)$ of a digraph $G$ is the smallest size of a locating-dominating set of $G$. We investigate upper bounds on $\gamma_L(G)$ 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 $\gamma_L(G)$ 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 $n$, $\gamma_L(G)\leq\frac{4n}{5}$ holds, and there exist twin-free digraphs $G$ with $\gamma_L(G)=\frac{2(n-2)}{3}$. If moreover $G$ is a tournament or is acyclic, the bound is improved to $\gamma_L(G)\leq\lceil\frac{n}{2}\rceil$, which is tight in both cases