On uniquely k-determined permutations

There are several approaches to study occurrences of consecutive patterns in permutations such as the inclusion-exclusion method, the tree representations of permutations, the spectral approach and others. We propose yet another approach to study occurrences of consecutive patterns in permutations. The approach is based on considering the graph of patterns overlaps, which is a certain subgraph of the de Bruijn graph. While applying our approach, the notion of a uniquely $k$-determined permutation appears. We give two criteria for a permutation to be uniquely $k$-determined: one in terms of the distance between two consecutive elements in a permutation, and the other one in terms of directed hamiltonian paths in the certain graphs called path-schemes. Moreover, we describe a finite set of prohibitions that gives the set of uniquely $k$-determined permutations. Those prohibitions make applying the transfer matrix method possible for determining the number of uniquely $k$-determined permutations.Comment: 12 page

Succinct Permutation Graphs

We present a succinct, i.e., asymptotically space-optimal, data structure for permutation graphs that supports distance, adjacency, neighborhood and shortest-path queries in optimal time; a variant of our data structure also supports degree queries in time independent of the neighborhood's size at the expense of an $O(\log n/\log \log n)$-factor overhead in all running times. We show how to generalize our data structure to the class of circular permutation graphs with asymptotically no extra space, while supporting the same queries in optimal time. Furthermore, we develop a similar compact data structure for the special case of bipartite permutation graphs and conjecture that it is succinct for this class. We demonstrate how to execute algorithms directly over our succinct representations for several combinatorial problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Moreover, we initiate the study of semi-local graph representations; a concept that "interpolates" between local labeling schemes and standard "centralized" data structures. We show how to turn some of our data structures into semi-local representations by storing only $O(n)$ bits of additional global information, beating the lower bound on distance labeling schemes for permutation graphs

Hamiltonian cycles in faulty random geometric networks

In this paper we analyze the Hamiltonian properties of faulty random networks. This consideration is of interest when considering wireless broadcast networks. A random geometric network is a graph whose vertices correspond to points uniformly and independently distributed in the unit square, and whose edges connect any pair of vertices if their distance is below some specified bound. A faulty random geometric network is a random geometric network whose vertices or edges fail at random. Algorithms to find Hamiltonian cycles in faulty random geometric networks are presented.Postprint (published version

Parameterized Algorithms for Modular-Width

It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.Comment: to appear in IPEC 2013. arXiv admin note: text overlap with arXiv:1304.5479 by other author

Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs

A graph $G$ is {\em matching-decyclable} if it has a matching $M$ such that $G-M$ is acyclic. Deciding whether $G$ is matching-decyclable is an NP-complete problem even if $G$ is 2-connected, planar, and subcubic. In this work we present results on matching-decyclability in the following classes: Hamiltonian subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian subcubic graphs we show that deciding matching-decyclability is NP-complete even if there are exactly two vertices of degree two. For chordal and distance-hereditary graphs, we present characterizations of matching-decyclability that lead to $O(n)$-time recognition algorithms

Some local--global phenomena in locally finite graphs

In this paper we present some results for a connected infinite graph $G$ with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of $G$. (For a vertex $w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of $G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is 2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$ has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017). Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's condition (1960) then all balls of any radius in $G$ are Hamiltonian.Comment: 18 pages, 6 figures; journal accepted versio