Asymptotic enumeration of non-crossing partitions on surfaces

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft

Connected and internal graph searching

This paper is concerned with the graph searching game. The search number es(G) of a graph G is the smallest number of searchers required to clear G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed. The difficulty of the connected version and of the monotone internal version of the graph searching problem comes from the fact that, as shown in the paper, none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of the graph searching problem. Motivated by the fact that connected graph searching, and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is only one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T). For arbitrary graphs, we prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, es(G)= is(G)= ms(G)leq mis(G)leq cs(G)= ics(G)leq mcs(G)=mics(G). The first two inequalities can be strict. In the case of trees, we have mics(G)leq 2 es(T)-2, that is there are exactly 2 different search numbers in trees, and these search numbers differ by a factor of 2 at most.Postprint (published version

Structure and enumeration of K4-minor-free links and link diagrams

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.Peer ReviewedPostprint (author's final draft

On the existence of Nash equilibria in strategic search games

We consider a general multi-agent framework in which a set of n agents are roaming a network where m valuable and sharable goods (resources, services, information ....) are hidden in m different vertices of the network. We analyze several strategic situations that arise in this setting by means of game theory. To do so, we introduce a class of strategic games that we call strategic search games. In those games agents have to select a simple path in the network that starts from a predetermined set of initial vertices. Depending on how the value of the retrieved goods is splitted among the agents, we consider two game types: finders-share in which the agents that find a good split among them the corresponding benefit and firsts-share in which only the agents that first find a good share the corresponding benefit. We show that finders-share games always have pure Nash equilibria (pne ). For obtaining this result, we introduce the notion of Nash-preserving reduction between strategic games. We show that finders-share games are Nash-reducible to single-source network congestion games. This is done through a series of Nash-preserving reductions. For firsts-share games we show the existence of games with and without pne. Furthermore, we identify some graph families in which the firsts-share game has always a pne that is computable in polynomial time.Peer ReviewedPostprint (author’s final draft

A list of parameterized problems in bioinformatics

In this report we present a list of problems that originated in bionformatics. Our aim is to collect information on such problems that have been analyzed from the point of view of Parameterized Complexity. For every problem we give its definition and biological motivation together with known complexity results.Postprint (published version

Contraction checking in graphs on surfaces

The CONTRACTION CHECKING problem asks, given two graphs H and G as input, whether H can be obtained from G by a sequence of edge contractions. CONTRACTION CHECKING remains NP-complete, even when H is fixed. We show that this is not the case when G is embeddable in a surface of fixed Euler genus. In particular, we give an algorithm that solves CONTRACTION CHECKING in f(h,g) ·SCOPUS: cp.pinfo:eu-repo/semantics/publishe

A Simple and fast approach for solving problems on planar graphs

It is well known that the celebrated Lipton-Tarjan planar separation theorem, in a combination with a divide-and-conquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2^{O(sqrt{n})}, where n is the number of vertices. However, the constants hidden in big-Oh, usually are too large to claim the algorithms to be practical even on graphs of moderate size. Here we introduce a new algorithm design paradigm for solving problems on planar graphs. The paradigm is so simple that it can be explained in any textbook on graph algorithms: Compute tree or branch decomposition of a planar graph and do dynamic programming. Surprisingly such a simple approach provides faster algorithms for many problems. For example, Independent Set on planar graphs can be solved in time O(2^{3.182sqrt{n}}n+n^4) and Dominating Set in time O(2^{5.043sqrt{n}}n+n^4). In addition, significantly broader class of problems can be attacked by this method. Thus with our approach, Longest cycle on planar graphs is solved in time O(2^{2.29sqrt{n}(ln{n}+0.94)}n^{5/4}+n^4) and Bisection is solved in time O(2^{3.182sqrt{n}}n+n^4). The proof of these results is based on complicated combinatorial arguments that make strong use of results derived by the Graph Minors Theory. In particular we prove that branch-width of a planar graph is at most 2.122sqrt{n}. In addition we observe how a similar approach can be used for solving different fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speed-up for fundamental problems on planar graphs like Vertex Cover, (Weighted) Dominating Set, and many others.Postprint (published version

Monotonicity and inert fugitive search games

In general, graph search can be described in terms of a sequence of searchers' moves on a graph, able to capture a fugitive resorting on its vertices/edges. Several variations of graph search have been introduced, differing on the abilities of the fugitive as well as of the search. In this paper, we examine the case where the fugitive is inert, i.e., it moves only whenever the search is about to capture it. Mainly, there are two variants fo

On the monotonicity of games generated by symmetric submodular functions

Submodular functions have appeared to be a key tool for proving the monotonicity of several graph searching games. In this paper we provide a general game theoretic framework able to unify old and new monotonicity results in a unique min-max theorem. Our theorem, provides a game theoretic analogue to a wide number of graph theoretic parameters such as linear-width and cutwidth

On the monotonicity of games generated by symmetric submodular functions

Submodular functions have appeared to be a key tool for proving the monotonicity of several graph searching games. In this paper we provide a general game theoretic framework able to unify old and new monotonicity results in a unique min-max theorem. Our theorem, provides a game theoretic analogue to a wide number of graph theoretic parameters such as linear-width and cutwidth