A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs

A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph

3-Colourability of Dually Chordal Graphs in Linear Time

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four

Local Certification of Some Geometric Intersection Graph Classes

In the context of distributed certification, the recognition of graph classes has started to be intensively studied. For instance, different results related to the recognition of planar, bounded tree-width and $H$-minor free graphs have been recently obtained. The goal of the present work is to design compact certificates for the local recognition of relevant geometric intersection graph classes, namely interval, chordal, circular arc, trapezoid and permutation. More precisely, we give proof labeling schemes recognizing each of these classes with logarithmic-sized certificates. We also provide tight logarithmic lower bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

Computing Graph Roots Without Short Cycles

Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H2 for some graph H of small girth. The main results are the following. - There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism. - There is a polynomial time algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is NP-complete to recognize if G = H2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed

Building Information Filtering Networks with Topological Constraints: Algorithms and Applications

We propose a new methodology for learning the structure of sparse networks from data; in doing so we adopt a dual perspective where we consider networks both as weighted graphs and as simplicial complexes. The proposed learning methodology belongs to the family of preferential attachment algorithms, where a network is extended by iteratively adding new vertices. In the conventional preferential attachment algorithm a new vertex is added to the network by adding a single edge to another existing vertex; in our approach a new vertex is added to a set of vertices by adding one or more new simplices to the simplicial complex. We propose the use of a score function to quantify the strength of the association between the new vertex and the attachment points. The methodology performs a greedy optimisation of the total score by selecting, at each step, the new vertex and the attachment points that maximise the gain in the score. Sparsity is enforced by restricting the space of the feasible configurations through the imposition of topological constraints on the candidate networks; the constraint is fulfilled by allowing only topological operations that are invariant with respect to the required property. For instance, if the topological constraint requires the constructed network to be be planar, then only planarity-invariant operations are allowed; if the constraint is that the network must be a clique forest, then only simplicial vertices can be added. At each step of the algorithm, the vertex to be added and the attachment points are those that provide the maximum increase in score while maintaining the topological constraints. As a concrete but general realisation we propose the clique forest as a possible topological structure for the representation of sparse networks, and we allow to specify further constraints such as the allowed range of clique sizes and the saturation of the attachment points. In this thesis we originally introduce the Maximally Filtered Clique Forest (MFCF) algorithm: the MFCF builds a clique forest by repeated application of a suitably invariant operation that we call Clique Expansion operator and adds vertices according to a strategy that greedily maximises the gain in a local score function. The gains produced by the Clique Expansion operator can be validated in a number of ways, including statistical testing, cross-validation or value thresholding. The algorithm does not prescribe a specific form for the gain function, but allows the use of any number of gain functions as long as they are consistent with the Clique Expansion operator. We describe several examples of gain functions suited to different problems. As a specific practical realisation we study the extraction of planar networks with the Triangulated Maximally Filtered Graph (TMFG). The TMFG, in its simplest form, is a specialised version of the MFCF, but it can be made more powerful by allowing the use of specialised planarity invariant operators that are not based on the Clique Expansion operator. We provide applications to two well known applied problems: the Maximum Weight Planar Subgraph Problem (MWPSP) and the Covariance Selection problem. With regards to the Covariance Selection problem we compare our results to the state of the art solution (the Graphical Lasso) and we highlight the benefits of our methodology. Finally, we study the geometry of clique trees as simplicial complexes and note how the statistics based on cliques and separators provides information equivalent to the one that can be achieved by means of homological methods, such as the analysis of Betti numbers, however with our approach being computationally more efficient and intuitively simpler. Finally, we use the geometric tools developed to provide a possible methodology for inferring the size of a dataset generated by a factor model. As an example we show that our tools provide a solution for inferring the size of a dataset generated by a factor model

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results