Section: New Results

Graph Theory

Participants : Julio Araújo, Jean-Claude Bermond, Frédéric Giroire, Frédéric Havet, František Kardoš, Ana Karolinna Maia, Remigiusz Modrzejewski, Leonardo Sampaio, Michel Syska.

Algorithms in graphs

Mascotte is also interested in the algorithmic aspects of Graph Theory. In general we try to find the most efficient algorithms to solve various problems of Graph Theory and telecommunication networks.

Complexity and Computation of Graph Parameters

We used graph theory to model various networks' problems. In general we study their complexity and then we investigate the structural properties of graphs that make these problems hard or easy. In particular, we try to find the most efficient algorithms to solve the problems, sometimes focusing on specific graph classes where the problems are polynomial-time solvable.

Degree Constraint Subgraphs. A natural question in current social networks is How do one find a small community (subgraph) in which anyone as at least $d$ friends (neighbors)? This problem can be modelled as degree-constrained subgraph problems where the objective is to find an optimal weighted subgraph, subject to certain degree constraints (in which each node has degree at most $d$), in a weighted graph. When $d=2$, the problem is easy to solve since one simply needs to compute the girth of the graph. In [16] , we proved that the problem is not in Apx for any $d\ge 3$. The proof is obtained by a reduction from Vertex Cover in regular graphs, followed by the use of an error amplification technique. On the positive side, we give an $\frac{n}{logn}$-approximation algorithm for the class of graphs excluding a fixed graph H as a minor (including planar or bounded genus graphs), using dynamic programming.

Hyperbolicity in Large graphs. Hyperbolicity is a geometric notion that measure how the various shortest paths connecting two vertices can diverge in a graph. Knowing its value provides information on the geometry of the network, moreover it has practical implications for shortest path routing. Hyperbolicity can be computed in polynomial time algorithm ($\Theta \left(n^4\right)$). This is far from being practical for large graphs. So, in [69] we proposed a scalable algorithm for this problem. We also led some computational experiments of our algorithms on large-scale graphs.

Hull Number of graphs. In [64] , we study the (geodesic) hull number of graphs. For any two vertices $u,v\in V$ of a connected undirected graph $G=(V,E)$, the closed interval $I[u,v]$ of $u$ and $v$ is the set of vertices that belong to some shortest $(u,v)$-path. For any $S\subseteq V$, let $I\left[S\right]={\bigcup }_{u,v\in S}I[u,v]$. A subset $S\subseteq V$ is (geodesically) convex if $I\left[S\right]=S$. Given a subset $S\subseteq V$, the convex hull $I_h\left[S\right]$ of $S$ is the smallest convex set that contains $S$. We say that $S$ is a hull set of $G$ if $I_h\left[S\right]=V$. The size of a minimum hull set of $G$ is the hull number of $G$, denoted by $hn\left(G\right)$. First, we show a polynomial-time algorithm to compute the hull number of any $P_5$-free triangle-free graph. Then, we present four reduction rules based on vertices with the same neighborhood. We use these reduction rules to propose a fixed parameter tractable algorithm to compute the hull number of any graph $G$, where the parameter can be the size of a vertex cover of $G$ or, more generally, its neighborhood diversity, and we also use these reductions to characterize the hull number of the lexicographic product of any two graphs. More on the hull number of graphs may be found in Araujo's thesis [13] .

Graph Searching, Cops and Robber Games

Pursuit-evasion encompasses a wide variety of combinatorial problems related to the capture of a fugitive residing in a network by a team of searchers. The goal consists in minimizing the number of searchers required to capture the fugitive in a network and in computing the corresponding capture strategy. This can also be viewed as cleaning the edges of a contaminated graph. We investigated several variants of these games.

Web Caching & the surfer Game. A surprising application of some variant of pursuit-evasion games (namely Cops and Robber games) is the problem for a web-browser to download documents in advance while an internaut is surfing on the Web. In [53] , [52] , we provide a modelling of the prefetching problem in terms of Cops and Robber games. The parameter to be optimized is then the download-speed necessary for the Internaut only accesses to already download webpages. This allows us to provide several complexity results and polynomial-time algorithms in some graph classes.

Connected Graph Searching. Another variant of pursuit-evasion games is graph searching which is mainly related to graph decompositions. For instance, the minimum number of searchers needed to capture an invisble fugitive in a graph is equal to its pathwidth plus one. In [21] , we investigated the connected variant of this game. A strategy is called connected if the clear part (the part where the fugitive cannot stand) always induces a connected subgraph. The main motivation for studying connected graph searching is the design of distributed protocols allowing searchers to compute a capture strategy (see also Section  6.2.1.3 ). [21] gathers most of the results of the last decade concerning connected graph searching, mainly focussing on the cost of connectivity in terms of number of searchers.

Distributed Algorithms

We investigated algorithmic problems arising in complex networks like the Internet or social networks. In this kind of networks, problems are becoming harder or impracticable because of the size and the dynamicity of these networks. One way to handle the dynamicity is to provide (distributed) fault tolerant algorithms. Studying the mobile agents paradigm seems to be a promissing approach (somehow related to Cops and Robber in Section  6.2.1.2 ) to adress some models of distributed computing. We considered distributed or even self-stabilizing algoritms for gathering and graph searching problems.

Graph Searching and Routing Reconfiguration. In [29] , we developed a generic distributed algorithm for computing and updating various parameters on trees including the process number (see Section  6.1.2.3 ), and other related graph searching parameters (see Section  6.2.1.2 ). We also proposed an incremental version of the algorithm allowing to update these parameters after addition or deletion of any tree edge.

Robots in anonymous networks. Motivated by the understanding of the limits of distributed computing, we consider a recent model of robot-based computing which makes use of identical, memoryless mobile robots placed on nodes of anonymous graphs. The robots operate in Look-Compute-Move cycles that are performed asynchronously for each robot. In particular, we consider various problems such as graph exploration, graph searching and gathering in various graph classes. We provide a new distributed approach which turns out to be very interesting as it neither completely falls into symmetry-breaking nor into symmetry-preserving techniques. More precisely, we design algorithms for the gathering in rings [51] , [70] , grid [50] and trees [61] . We also proposed a general approach [71] to solve the three problems in rings. Finally, in [67] , [44] , [43] , algorithms are designed to solve the graph searching problem in trees.

Structural graph theory

Directed graphs

Graph theory can be roughly partitioned into two branches: the areas of undirected graphs and directed graphs (digraphs). Even though both areas have numerous important applications, for various reasons, undirected graphs have been studied much more extensively than directed graphs. One of the reasons is that many problems for digraphs are much more difficult than their analogues for directed graphs. For example, one of the cornerstones of modern (undirected) graph theory is Minor Theory of Robertson and Seymour. Unfortunately, we cannot expect an equivalent for directed graphs. Minor Theory implies in particular that, for any fixed $H$, detecting a subdivision of $H$ in an input graph $G$ can be performed in polynomial time by the Robertson and Seymour linkage algorithm. In contrast, the analogous subdivision problem for digraph can be either polynomial-time solvable or NP-complete, depending on the fixed digraph $H$. In [65] , we give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems. We also investigated the related problem in which we want to detect an induced subdivision of $H$. Already, for undirected graphs the complexity of this problem depends on $H$. In [20] , we show that for digraphs the complexity of this problem depends on $H$ and on whether the input digraph $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs.

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In [34] , we consider the Spanning Galaxy problem of deciding whether a digraph $D$ has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynomial-time solvable when restricted to strong digraphs. In fact, we prove that restricted to this class, the Spanning Galaxy problem is equivalent to the problem of deciding if a strong digraph has a strong digraph with an even number of vertices. We then show a polynomial-time algorithm to solve this problem. We also consider some parameterized versions of the Spanning Galaxy problem. Finally, we improve some results concerning the notion of directed star arboricity of a digraph $D$, denoted $dst\left(D\right)$, which is the minimum number of galaxies needed to cover all the arcs of $D$. We show in particular that $dst\left(D\right)\le \Delta \left(D\right)+1$ for every digraph $D$ and that $dst\left(D\right)\le \Delta \left(D\right)$ for every acyclic digraph $D$.

Hypergraphs are a generalization of graphs, in which every edge is incident to a set of vertices of any size (not necessarily 2). Like for digraphs, a lot fewer is known about them than about graphs. The two notions of eulerian and hamoltinians cycles have been extensively studied for graphs and digraphs. The analogue notion of eulerian cycle in a hypergraph was only introduced in 2010 by Lonc and Naroski. In [72] , we introduce the notions of eulerian and hamiltonian circuits in directed hypergraphs. We show that both associated decision problems are NP-complete. Some necessary conditions for a dihypergraph to be have an eulerian circuit are presented. We exhibit some families of hypergraphs for which those are sufficient conditions. We also generalize a part of the properties of eulerian digraphs to the uniform and regular directed hypergraphs. Finally, we show that the de Bruijn and Kautz dihypergraphs are eulerian and hamiltonian in most cases.

Graph colouring

We mainly study graph colouring problems that model channel assignment problems.

A well-known such general problem is the following: we are given a graph $G$, whose vertices correspond to transmitters, together with an edge-weghting $w$. The weight of an edge corresponds to the minimum separation between the channels on its endvertices to avoid interferences. (If there is no edge, no separation is required, the transmitters do not interfere.) We need to assign positive integers (corresponding to channels) to the vertices so that for every edge $e$ the channels assigned to its endvertices differ by at least $w\left(e\right)$. The goal is to minimize the largest integer used, which corresponds to minimizing the span of the used bandwidth.

We mainly studied a particular, yet quite general, case, called backbone colouring, in which there are only two levels of interference. So we are given a graph $G$ and a subgraph $H$, called the backone. Two adjacent vertices in $H$ must get integers at least $q$ apart, while adjacent vertices in $G$ must get integers at distance at least 1. The minimum span is this case is called the $q$-backbone chromatic number and is denoted $BB{C}_q(G,H)$. Backbone forests in planar graphs are of particular interests. In [74] , we give a series of NP-hardness results as well as upper bounds for $BB{C}_q(G,H)$, depending on the type of the forest (matching, galaxy, spanning tree). Eventually, we discuss a circular version of the problem. In [73] , we also consider a list version of the problem in which every vertex must be assigned an integer in its own list of available ones. We provide bounds using the Combinatorial Nullstellensatz for the list version on the channel assignment problem. Through this result and through structural approaches, we obtain good upper bounds for forests and matching backbone in planar graphs. In [68] , we give an evidence to a conjecture of Broersma et al. stating that $BB{C}_2(G,T)\le 6$, for every planar graph $G$ and spanning tree $T$. We prove this conjecture in the particular case when $T$ has diameter at most 4.

Another meaningful and very well-studied particular case of backbone colouring is $L(p,1)$-labelling, which is $p$-backbone colouring of $(G^2,G)$, where $G^2$ is the square of $G$ (the graph with same vertex set as $G$, in which two vertices are adjacents if they are at distance at most 2 in $G$). Griggs and Yeh conjecture in 1992, that for every graph with maximum degree $\Delta \ge 2$, $BB{C}_2(G^2,G)\le {\Delta }^2+1$. In [36] , we prove this conjecture when $\Delta$ is large. In fact, we prove a more general statement. We prove for any $q$ and sufficiently large $\Delta$, if $\Delta \left(H\right)\le {\Delta }^2$ and $\Delta \left(G\right)\le \Delta$, then $BB{C}_q(H,G)\le {\Delta }^2+1$. Our result also holds for the list version.

In [17] , we studied another colouring problem motivated by a practical frequency assignment problem and, up to our best knowledge, new. In wireless networks, a node interferes with other nodes, the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles,... We model this with a weighted graph $(G,w)$ where the weight function $w$ on the edges of $G$ represents the noise (interference) between the two end-vertices. The total interference in a node is the sum of all the noises of the nodes emitting on the same frequency. A weighted $t$-improper $k$-colouring of $(G,w)$ is a $k$-colouring of the nodes of $G$ (assignment of $k$ frequencies) such that the interference at each node does not exceed the threshold $t$. We consider the Weighted Improper Colouring problem which consists in determining the weighted $t$-improper chromatic number defined as the minimum integer $k$ such that $(G,w)$ admits a weighted $t$-improper $k$-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where, given a number $k$ of colours, we want to determine the minimum real $t$ such that $(G,w)$ admits a weighted $t$-improper $k$-colouring. We show that both problems are NP-hard and present general upper bounds for both problems; in particular we show a generalisation of Lovász's Theorem for the weighted $t$-improper chromatic number. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes at distance two. We derive the weighted $t$-improper chromatic number for all values of $t$.

Since some of the channel assignment problems must be done on-line, we are interested in some on-line graph colouring heuristics. We only studied such heuristics for the classical proper colouring. The easiest one, and the most widespread one, is the greedy algorithm, which colours the vertices one after another, giving to each vertex the smallest possible positive integer that is not already used by one of its neighbours. The Grundy number of a graph $G$ is the largest number of colours used by any execution of the greedy algorithm to colour $G$. In [27] , we give new bounds on the Grundy number of the different product of two graphs. The problem of determining the Grundy number of $G$ is polynomial-time solvable if $G$ is a $P_4$-free graph and NP-hard if $G$ is a $P_5$-free graph. In [19] , we define a new class of graphs, the fat-extended $P_4$-laden graphs, and we show a polynomial-time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of $P_5$-free graphs and strictly contains the class of $P_4$-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: $P_4$-reducible, extended $P_4$-reducible, $P_4$-sparse, extended $P_4$-sparse, $P_4$-extendible, $P_4$-lite, $P_4$-tidy, $P_4$-laden and extended $P_4$-laden, which are all strictly contained in the fat-extended $P_4$-laden class.

A colouring $c$ of a graph $G=(V,E)$ is a $b$-colouring if in every colour class there is a vertex whose neighborhood intersects every other colour classes. Such a colouring appears, when we try to optimize on-line the colouring of a graph, by changing the colour of all vertices of a colour class if it is possible. The $b$-chromatic number of $G$, denoted ${\chi }_b\left(G\right)$, is the greatest integer $k$ such that $G$ admits a $b$-coloring with $k$ colours. A graph $G$ is tight if it has exactly $m\left(G\right)$ vertices of degree $m\left(G\right)-1$, where $m\left(G\right)$ is the largest integer $m$ such that $G$ has at least $m$ vertices of degree at least $m-1$. Determining the $b$-chromatic number of a tight graph had been shown to be NP-hard even for a connected bipartite graph. In [35] , we show that it is also NP-hard for a tight chordal graph, and that the $b$-chromatic number of a split graph can be computed in polynomial time. Then we define the $b$-closure and the partial $b$-closure of a tight graph, and use these concepts to give a characterization of tight graphs whose $b$-chromatic number is equal to $m\left(G\right)$. This characterization is used to develop polynomial-time algorithms for deciding whether ${\chi }_b\left(G\right)=m\left(G\right)$, for tight graphs that are complement of bipartite graphs, $P_4$-sparse and block graphs. We generalize the concept of pivoted tree introduced by Irving and Manlove and show its relation with the $b$-chromatic number of tight graphs.

Many more results on greedy colourings and $b$-colourings have been proved in Sampaio's thesis [14] .

We studied other variations of graph colouring. In [18] , we aim at characterizing the class of graphs that admit a good edge-labelling. Such graphs are interesting, as they correspond to set of requests in UPP-digraphs (in which there is at most one dipath from a vertex to another) for which the minimum number of wavelengths is equal to the maximum load. This implies that the problem can be solved efficiently. First, we exhibit infinite families of graphs for which no good edge-labelling can be found. We then show that deciding if a graph admits a good edge-labelling is NP-complete. Finally, we give large classes of graphs admitting a good edge-labelling: $C_3$-free outerplanar graphs, planar graphs of girth at least 6, subcubic $\{C_3,K_{2,3}\}$-free graphs.

For a connected graph $G$ of order at least 3 and a $k$-labelling $c:E\left(G\right)\to \{1,2,\cdots ,k\}$ of the edges of $G$, the code of a vertex $v$ of $G$ is the ordered $k$-tuple $(n_1,\cdots ,n_k)$, where $n_i$ is the number of edges incident with $v$ that are labelled $i$. The $k$-labelling $c$ is detectable if every two adjacent vertices of $G$ have distinct codes. The minimum positive integer $k$ for which $G$ has a detectable $k$-labelling is the detection number of $G$. In [76] , we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.