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1. Recognition and Drawing of Stick Graphs

   https://doi.org/10.1007/978-3-030-04414-5_21  

   De Luca, F.; Hossain, I.; Kobourov, S.; Lubiw, A.; Mondal, D. (January 2018, International Symposium on Graph Drawing and Network Visualization)

   A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a “ground line,” a line with slope −1. It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G’s bipartite adjacency matrix with rows A and columns B excludes three small ‘forbidden’ submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A|3|B|3)-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable. 

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2. Visualization of Bipartite Graphs in Limited Window Size

   https://doi.org/10.1007/978-3-031-52113-3_14  

   Evans, William; Koeck, Kassian; Kobourov, Stephen (February 2024, 49th Intl. Conference on Current Trends in Theory and Practice of Computer Science (SofSem))

   Bipartite graphs are commonly used to visualize objects and their features. An object may possess several features and several objects may share a common feature. The standard visualization of bipartite graphs, with objects and features on two (say horizontal) parallel lines at integer coordinates and edges drawn as line segments, can often be difficult to work with. A common task in visualization of such graphs is to consider one object and all its features. This naturally defines a drawing window, defined as the smallest interval that contains the x-coordinates of the object and all its features. We show that if both objects and features can be reordered, minimizing the average window size is NP-hard. However, if the features are fixed, then we provide an efficient polynomial time algorithm for arranging the objects, so as to minimize the average window size. Finally, we introduce a different way of visualizing the bipartite graph, by placing the nodes of the two parts on two con- centric circles. For this setting we also show NP-hardness for the general case and a polynomial time algorithm when the features are fixed. 

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3. Enumeration of Intersection Graphs of x-Monotone Curves

   https://doi.org/10.4230/LIPIcs.GD.2024.4  

   Fox, Jacob; Pach, János; Suk, Andrew (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Felsner, Stefan; Klein, Karsten (Ed.)

   A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^Ω(n^{4/3}) families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O(n^{3/2-ε}), where ε > 0 is a suitable constant. Our proof uses an upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number. 

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4. A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

   Lahn, Nathaniel; Raghvendra, Sharath (June 2019, Leibniz international proceedings in informatics)

   We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm. 

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5. A Faster Algorithm for Minimum-cost Bipartite Perfect Matching in Planar Graphs

   https://doi.org/10.1145/3365006  

   Asathulla, Mudabir Kabir; Khanna, Sanjeev; Lahn, Nathaniel; Raghvendra, Sharath (January 2020, ACM Transactions on Algorithms)

   null (Ed.)

   Given a weighted planar bipartite graph G ( A ∪ B , E ) where each edge has an integer edge cost, we give an Õ( n 4/3 log nC ) time algorithm to compute minimum-cost perfect matching; here C is the maximum edge cost in the graph. The previous best-known planarity exploiting algorithm has a running time of O ( n 3/2 log n ) and is achieved by using planar separators (Lipton and Tarjan ’80). Our algorithm is based on the bit-scaling paradigm (Gabow and Tarjan ’89). For each scale, our algorithm first executes O ( n 1/3 ) iterations of Gabow and Tarjan’s algorithm in O ( n 4/3 ) time leaving only O ( n 2/3 ) vertices unmatched. Next, it constructs a compressed residual graph H with O ( n 2/3 ) vertices and O ( n ) edges. This is achieved by using an r -division of the planar graph G with r = n 2/3 . For each partition of the r -division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortest-path data structures, the remaining O ( n 2/3 ) vertices are matched by iteratively computing a minimum-cost augmenting path, each taking Õ( n 2/3 ) time. Augmentation changes the residual graph, so the algorithm updates the compressed representation for each partition affected by the change in Õ( n 2/3 ) time. We bound the total number of affected partitions over all the augmenting paths by O ( n 2/3 log n ). Therefore, the total time taken by the algorithm is Õ( n 4/3 ). 

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