A note on circular chromatic number of graphs with large girth and similar problems

In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most $5$, and for the $p$th chromatic number $\chi_p$, from which follows in particular that graphs $G$ of large girth excluding a minor have $\chi_p(G)\leq p+2$

Preventing Unraveling in Social Networks Gets Harder

The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree $k$ the equilibrium for this model is a maximal subgraph whose minimum degree is $\geq k$. However the dropping out of individuals with degrees less than $k$ might lead to a cascading effect of iterated withdrawals such that the size of equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored $k$-Core} problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and every vertex $v\in H\setminus B$ has degree at least $k$ is the induced subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ and gave some inapproximability and fixed-parameter intractability results. In this paper we give improved hardness results for this problem. In particular we show that the \textsc{Anchored $k$-Core} problem is W[1]-hard parameterized by $p$, even for $k=3$. This improves the result of Bhawalkar et al. \cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our parameter is always bigger since $p\geq b$. Then we answer a question of Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored $k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if the maximum degree of the graph is $k+2$. Finally we show that the problem is FPT on planar graphs parameterized by $b$ for all $k\geq 7$.Comment: To appear in AAAI 201

Recovering sparse graphs

We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d,k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in classes of sparse graphs. We derive as a corollary that if G_0 is a graph class with bounded expansion, then the first order model checking is fixed parameter tractable in the class of all graphs that can obtained from a graph G from G_0 by complementing on at most k arbitrary subsets of the vertex set of G; this implies an earlier result that the first order model checking is fixed parameter tractable in graph classes interpretable in classes of graphs with bounded maximum degree

Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries

On the Monadic Second-Order Transduction Hierarchy

We compare classes of finite relational structures via monadic second-order transductions. More precisely, we study the preorder where we set C \subseteq K if, and only if, there exists a transduction {\tau} such that C\subseteq{\tau}(K). If we only consider classes of incidence structures we can completely describe the resulting hierarchy. It is linear of order type {\omega}+3. Each level can be characterised in terms of a suitable variant of tree-width. Canonical representatives of the various levels are: the class of all trees of height n, for each n \in N, of all paths, of all trees, and of all grids

Testing first-order properties for subclasses of sparse graphs

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nesetril and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings