A separator-based method for generating weakly chordal graphs

We propose a scheme for generating a weakly chordal graph on n vertices with m edges. In this method, we first construct a tree and then generate an orthogonal layout (which is a weakly chordal graph on the n vertices) based on this tree. In the next and final step, we insert additional edges to give us a weakly chordal graph on m edges. Our algorithm ensures that the graph remains weakly chordal after each edge is inserted. The time complexity of an insertion query is O(n^3) time and an insertion takes constant time. On the other hand, a generation algorithm based on finding a 2-pair takes O(nm) time using the algorithm of Arikati and Rangan [1].Comment: 14 pages and 30 figure

Generating Weakly Chordal Graphs from Arbitrary Graphs

We propose a scheme for generating a weakly chordal graph from a randomly generated input graph, G = (V, E). We reduce G to a chordal graph H by adding fill-edges, using the minimum vertex degree heuristic. Since H is necessarily a weakly chordal graph, we use an algorithm for deleting edges from a weakly chordal graph that preserves the weak chordality property of H. The edges that are candidates for deletion are the fill-edges that were inserted into G. In order to delete a maximal number of fill-edges, we maintain these in a queue. A fill-edge is removed from the front of the queue, which we then try to delete from H. If this violates the weak chordality property of H, we reinsert this edge at the back of the queue. This loop continues till no more fill-edges can be removed from H. Operationally, we implement this by defining a deletion round as one in which the edge at the back of the queue is at the front.We stop when the size of the queue does not change over two successive deletion rounds and output H.Comment: 15 pages, 29 figure

Sum-perfect graphs

Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the stability number and clique number, respectively.) We give a set of $27$ graphs and we prove that a graph $G$ is sum-perfect if and only if $G$ does not contain any of the graphs in the set as an induced subgraph.Comment: 10 pages, 3 figure

Transitive orientations in bull-reducible Berge graphs

A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We show that for each $k \geq 3$ there is a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colourable compact graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all $k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5, \overline{P_5}, C_5)$-free graphs. We also mention some open problems.Comment: 9 pages, 4 figures; minor change

Multi-View Optimization of Local Feature Geometry

In this work, we address the problem of refining the geometry of local image features from multiple views without known scene or camera geometry. Current approaches to local feature detection are inherently limited in their keypoint localization accuracy because they only operate on a single view. This limitation has a negative impact on downstream tasks such as Structure-from-Motion, where inaccurate keypoints lead to large errors in triangulation and camera localization. Our proposed method naturally complements the traditional feature extraction and matching paradigm. We first estimate local geometric transformations between tentative matches and then optimize the keypoint locations over multiple views jointly according to a non-linear least squares formulation. Throughout a variety of experiments, we show that our method consistently improves the triangulation and camera localization performance for both hand-crafted and learned local features.Comment: Accepted at ECCV 2020. 28 pages, 11 figures, 6 table

Weakly Chordal Graphs: An Experimental Study

Graph theory is an important field that enables one to get general ideas about graphs and their properties. There are many situations (such as generating all linear layouts of weakly chordal graphs) where we want to generate instances to test algorithms for weakly chordal graphs. In my thesis, we address the algorithmic problem of generating weakly chordal graphs. A graph G=(V, E), where V is its vertices and E is its edges, is called a weakly chordal graph, if neither G nor its complement G\u27, contains an induced chordless cycle on five or more vertices. Our work is in two parts. In the first part, we carry out a comparative study of two existing algorithms for generating weakly chordal graphs. The first algorithm for generating weakly chordal graphs repeatedly finds a two-pair and adds an edge between them. The second-generation algorithm starts by constructing a tree and then generates an orthogonal layout (also weakly chordal graph) based on this tree. Edges are then inserted into this orthogonal layout until there are $m$ edges. The output graphs from these two methods are compared with respect to several parameters like the number of four cycles, run times, chromatic number, the number of non-two-pairs in the graphs generated by the second method. In the second part, we propose an algorithm for generating weakly chordal graphs by edge deletions starting from an arbitrary input random graph. The algorithm starts with an arbitrary graph to be able to generate a weakly chordal graph by the basis of edge deletion. The algorithm iterates by maintaining weak chordality by preventing any hole or antihole configurations being formed for any successful deletion of an edge

A characterization of b-perfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number of a graph $G$ is the largest integer $k$ such that $G$ admits a b-coloring with $k$ colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph of $G$. We prove that a graph is b-perfect if and only if it does not contain as an induced subgraph a member of a certain list of twenty-two graphs. This entails the existence of a polynomial-time recognition algorithm and of a polynomial-time algorithm for coloring exactly the vertices of every b-perfect graph