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We study the PSPACE-complete $k$-Canadian Traveller Problem, where a weighted graph $G=(V,E,\omega)$ with a source $s\in V$ and a target $t\in V$ are given. This problem also has a hidden input $E_* \subsetneq E$ of cardinality at most $k$ representing blocked edges. The objective is to travel from $s$ to $t$ with the minimum distance. At the beginning of the walk, the blockages $E_*$ are unknown: the traveller discovers that an edge is blocked when visiting one of its endpoints. Online algorithms, also called strategies, have been proposed for this problem and assessed with the competitive ratio, i.e. the ratio between the distance actually traversed by the traveller divided by the distance we would have traversed knowing the blockages in advance. Even though the optimal competitive ratio is $2k+1$ even on unit-weighted planar graphs of treewidth 2, we design a polynomial-time strategy achieving competitive ratio $9$ on unit-weighted outerplanar graphs. This value $9$ also stands as a lower bound for this family of graphs as we prove that, for any $\varepsilon > 0$, no strategy can achieve a competitive ratio $9-\varepsilon$. Finally, we show that it is not possible to achieve a constant competitive ratio (independent of $G$ and $k$) on weighted outerplanar graphs.

We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every $3$-critical signed graph on $n$ vertices has at least $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least $6$ is (circular) $3$-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph $C_{3}^*$, which is the positive triangle augmented with a negative loop on each vertex., Comment: 27 pages, 12 figures

A graph database is a digraph whose arcs are labeled with symbols from a fixed alphabet. A regular graph pattern (RGP) is a digraph whose edges are labeled with regular expressions over the alphabet. RGPs model navigational queries for graph databases called conjunctive regular path queries (CRPQs). A match of a CRPQ in the database is witnessed by a special navigational homomorphism of the corresponding RGP to the database. We study the complexity of deciding the existence of a homomorphism between two RGPs. Such homomorphisms model a strong type of containment between the two corresponding CRPQs. We show that this problem can be solved by an EXPTIME algorithm (while general query containmement in this context is EXPSPACE-complete). We also study the problem for restricted RGPs over a unary alphabet, that arise from some applications like XPath or SPARQL. For this case, homomorphism-based CRPQ containment is in NP. We prove that certain interesting cases are in fact polynomial-time solvable., Comment: 15 pages. Short version appeared in the proceedings of the 15th Conference on Computability in Europe (CIE 2019)

The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours the vertices in the order of $\sigma$ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain {\em connected greedy colourings}. For some graphs, all connected greedy colourings use exactly $\chi(G)$ colours; they are called {\em good graphs}. On the opposite, some graphs do not admit any connected greedy colouring using only $\chi(G)$ colours; they are called {\em ugly graphs}. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no $K_4$-minor free graph is ugly. Moreover, our proofs are constructive, and imply the existence of polynomial-time algorithms to compute good connected orderings for these graph classes., Comment: 14 pages, 5 figures

A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture remains open even for graph metrics. In this article we prove that graphs with no induced house nor induced cycle of length at least~5 verify the desired property. We focus on lines generated by vertices at distance at most 2, define a new notion of ``good pairs'' that might have application in larger families, and finally use a discharging technique to count lines in irreducible graphs., Comment: 16 pages

A signed graph $(G, \Sigma)$ is a graph $G$ and a subset $\Sigma$ of its edges which corresponds to an assignment of signs to the edges: edges in $\Sigma$ are negative while edges not in $\Sigma$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,\Sigma)$ to $(H,\Pi)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipartite graph $(B, \Pi)$ of unbalanced-girth $2k$, we give a necessary and sufficient condition for $(B, \Pi)$ to admit a homomorphism from any signed bipartite graph of unbalanced-girth at least $2k$ whose underlying graph is $K_4$-minor-free. The condition can be checked in polynomial time with respect to the order of $B$. Let $SPC(2k)$ be the signed bipartite graph on vertex set $\mathbb{Z}_2^{2k-1}$ where vertices $u$ and $v$ are adjacent with a positive edge if their difference is in $\{e_1,e_2, \ldots, e_{2k-1}\}$ (where the $e_i$'s form the standard basis), and adjacent with a negative edge if their difference is $J$ (that is, the all-1 vector). As an application of our work, we prove that every signed bipartite $K_4$-minor-free graph of unbalanced-girth $2k$ admits a homomorphism to $SPC(2k)$. This supports a conjecture of Guenin claiming that every signed bipartite planar graph of unbalanced-girth $2k$ admits a homomorphism to $SPC(2k)$ (this would be an extension of the four-color theorem). We also give an application of our work to edge-coloring $2k$-regular $K_4$-minor-free multigraphs.

In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the Chen-Chv\'atal conjecture: their metric space (in the usual sense) has a universal line (in an unusual sense) or at least as many lines as the number of vertices.

We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of 1.4977 |V | and an upper bound of 1.5012 |V | .

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph $G$ is denoted $\gamma_b(G)$. The dual of this problem is called multipacking: a multipacking is a set $M$ of vertices such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$ . The maximum size of a multipacking in a graph $G$ is denoted mp(G). Naturally mp(G) $\leq \gamma_b(G)$. Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal.

The dual concepts of coverings and packings are well studied in graph theory. Coverings of graphs with balls of radius one and packings of vertices with pairwise distances at least two are the well-known concepts of domination and independence, respectively. In 2001, Erwin introduced \emph{broadcast domination} in graphs, a covering problem using balls of various radii, where the cost of a ball is its radius. The minimum cost of a dominating broadcast in a graph $G$ is denoted by $\gamma_b(G)$. The dual (in the sense of linear programming) of broadcast domination is \emph{multipacking}: a multipacking is a set $P \subseteq V(G)$ such that for any vertex $v$ and any positive integer $r$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $P$. The maximum size of a multipacking in a graph $G$ is denoted by $mp(G)$. Naturally, $mp(G) \leq \gamma_b(G)$. Hartnell and Mynhardt proved that $\gamma_b(G) \leq 3 mp(G) - 2$ (whenever $mp(G)\geq 2$). In this paper, we show that $\gamma_b(G) \leq 2mp(G) + 3$. Moreover, we conjecture that this can be improved to $\gamma_b(G) \leq 2mp(G)$ (which would be sharp)., Comment: 11 pages; 3 figures

Bousquet, Lochet and Thomass\'e recently gave an elegant proof that for any integer $n$, there is a least integer $f(n)$ such that any tournament whose arcs are coloured with $n$ colours contains a subset of vertices $S$ of size $f(n)$ with the property that any vertex not in $S$ admits a monochromatic path to some vertex of $S$. In this note we provide a lower bound on the value $f(n)$.

Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path P\_n is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars.

We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every $K_4$-minor free graph of odd-girth $2k+1$ admits a homomorphism to the projective hypercube of dimension $2k$. This supports a conjecture of the third author which generalizes the four-color theorem and relates to several outstanding conjectures such as Seymour's conjecture on edge-colorings of planar graphs. Strengthening this result, we show that the Kneser graph $K(2k+1,k)$ satisfies the conditions, thus implying that every $K_4$-minor free graph of odd-girth $2k+1$ has fractional chromatic number exactly $2+\frac{1}{k}$. Knowing that a smallest bound of odd-girth $2k+1$ must have at least ${k+2 \choose 2}$ vertices, we build nearly optimal bounds of order $4k^2$. Furthermore, we conjecture that the suprema of the fractional and circular chromatic numbers for $K_4$-minor-free graphs of odd-girth $2k+1$ are achieved by a same bound of odd-girth $2k+1$. If true, this improves, in the homomorphism order, earlier tight results on the circular chromatic number of $K_4$-minor-free graphs. We support our conjecture by proving it for the first few cases. Finally, as an application of our work, and after noting that Seymour provided a formula for calculating the edge-chromatic number of $K_4$-minor-free multigraphs, we show that stronger results can be obtained in the case of $K_4$-minor-free regular multigraphs., Comment: 27 pages, 10 figures

Given a graph $H$, a graph $G$ is called $H$-critical if $G$ does not admit a homomorphism to $H$, but any proper subgraph of $G$ does. Observe that $K_{k-1}$-critical graphs are the standard $k$-(colour)-critical graphs. We consider questions of extremal nature previously studied for $k$-critical graphs and generalize them to $H$-critical graphs. After complete graphs, the next natural case to consider for $H$ is that of the odd-cycles. Thus, given integers $\ell$ and $k$, $\ell\geq k$, we ask: what is the smallest order of a $C_{2\ell +1}$-critical graph of odd-girth at least $2k+1$? Denoting this value by $\eta(k,C_{2\ell+1})$, we show that $\eta(k,C_{2\ell+1})=4k$ for $1\leq\ell\leq k\leq\frac{3\ell+i-3}{2}$ ($2k=i\bmod 3$) and that $\eta(3,C_5)=15$. The latter means that a smallest graph of odd-girth~$7$ not admitting a homomorphism to the $5$-cycle is of order~$15$. Computational work shows that there are exactly eleven such graphs on $15$~vertices of which only two are $C_5$-critical., Comment: 16 pages, 8 figures. This version improves upon previous ones and uses the terminology of H-critical graphs, including a slight change of title

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 Thomass\'e) 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 $r$., Comment: 15 pages, 2 figures

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.

Given a directed graph D = (V,A) we define its intersection graph I(D) = (A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call these graphs facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize and they are easy to recognize when the graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problems on that class.

A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.

A graph is crossing-critical if the removal of any of its edges decreases its crossing number. This work is motivated by the following question: to what extent is crossing- criticality a property that is inherent to the structure of a graph, and to what extent can it be induced on a noncritical graph by multiplying (all or some of) its edges? It is shown that if a nonplanar graph G is obtained by adding an edge to a cubic polyhedral graph, and G is sufficiently connected, then G can be made crossing-critical by a suitable multiplication of edges., Comment: 12 pages - last proof updated

One of the De Bruijn - Erdos theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvatal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u,v,w} is a hyperedge. With this definition, the De Bruijn - Erdos theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result eneralizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.

We prove that two disjoint graphs must always be drawn separately on the Klein bottle, in order to minimize the crossing number of the whole drawing., Comment: 13 pages, second version, major changes in the proof