We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is p...

We prove the computational intractability of rotating and placing n square tiles into a 1 × n array such that adjacent tiles are compatible—either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NPhard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying th...

A geometric graph is a graph whose vertex set is a set of points in the plane and whose edge set contains straight-line segments. A matching in a graph is a subset of edges of the graph with no shared vertices. A matching is called perfect if it matches all the vertices of the underling graph. A geometric matching is a matching in a geometric graph. In this thesis, we study matching problems in...

We prove the computational intractability of rotating and placing n square tiles into a 1×n array such that adjacent tiles are compatible—either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the...

We study combinatorial optimisation problems on graphs in the mean-field model, which assigns independent and identically distributed random weights to the edges of the graph. Specifically, we focus on two generalisations of minimum weight matching on graphs. The first problem of minimum cost edge cover finds application in a computational linguistics problem of semantic projection. The second ...

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabă, Gregory, and Haemers. ...

We describe a system for measuring the angle of eye gaze error on people with a squint. The program to analyse an eye image models each eye simply as a circular iris within a sphere rotating about its centre. A combination of edge detection, image transformation and template matching are then used to best fit the model to the image of the eye. The algorithm behaves correctly on 95% of eye image...

We show that in any edge-coloring of the complete graph Kn on n vertices, such that each color class forms a complete bipartite graph, there is a spanning tree of Kn no two of whose edges have the same color. This strengthens a theorem of Graham and Pollak and verifies a conjecture of de Caen. More generally we show that in any edge-coloring of a graph G with p positive and q negative eigenvalu...

5 Given n red and n blue points in general position in the plane, it is well-known 6 that there is a perfect matching formed by non-crossing line segments. We charac-7 terize the bichromatic point sets which admit exactly one non-crossing matching. 8 We give several geometric descriptions of such sets, and find an O(n log n) algorithm 9 that checks whether a given bichromatic set has this prope...

Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of ...