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We abstract and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph G=(V,E) and a set of demand pairs P⊆VxV, a reachability preserver is a sparse subgraph H that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an n-node graph and demand pairs of the form P⊆SxV for a small node subset S, there is always a reachability preserver on O(n+√n/P//S/) edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which O(n) size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0.6+ε) of Chlamatac, et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534-443] (n4/7+ε). [ABSTRACT FROM AUTHOR]

Let k be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least 2k 1 contains k vertex-disjoint cycles. This conjecture is famous as one of a hundred unsolved problems selected in [A. Bondy and M. R. Murty, Graph Theory, Springer-Verlag, London, 2008]. Lichiardopol, P\'or, and Sereni proved in [SIAM J. Discrete Math., 23 (2009), pp. 979--992] that the above conjecture holds for k = 3. Let g be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy, and Thomass\'e conjectured in [J. Graph Theory, 75 (2014), pp. 284--302] that every digraph with girth g and minimum outdegree at least g g 1 k contains k vertex-disjoint cycles. Thomass\'e conjectured around 2006 that every oriented graph (a digraph without 2-cycles) with girth g and minimum outdegree at least h contains a path of length h(g 1), where h is a positive integer. In this paper, we first present a new shorter proof of the Bermond--Thomassen conjecture for the case of k = 3, and then we disprove the conjecture proposed by Bang-Jensen, Bessy, and Thomass\'e. Finally, we disprove the even girth case of the conjecture proposed by Thomass\'e. [ABSTRACT FROM AUTHOR]

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg [SIAM J. Discrete Math., 5 (1992), pp. 398--412] states that the queue number of planar graphs is bounded. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has bounded track number, and that every k-planar graph (i.e., every graph that can be drawn in the plane with at most k crossings per edge) of bounded degree has bounded queue number. [ABSTRACT FROM AUTHOR]

A graph G is called (2,k)‐connected if G is 2k‐edge‐connected and G−v is k‐edge‐connected for every vertex v. The study of (2,2)‐connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a 2‐vertex‐connected orientation if and only if it is (2,2)‐connected. In this paper, we provide a construction of the family of (2,k)‐connected graphs for k even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)‐connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph G and an integer k≥2, what is the minimum number of edges to be added to make G(2,k)‐connected. Both these results are based on a new splitting‐off theorem for (2,k)‐connected graphs. [ABSTRACT FROM AUTHOR]

Online social networks have continued to attract increased attention since the introduction of this concept nearly three decades ago. Consequently, a study about the workings of online social networks may help in understanding the structure of human society and the characteristics of generic complex networks. Over the past few years, interest in neighboring nodes, which are the number of nodes between any two nodes, the number of neighbors and how many triangles are present in a social network have produced the concepts of a six-degree separation (Milgram, Psychol Today 2(1):60–67, 1967; Backstrom et al., Proceedings of the 4th annual ACM web science conference, p 33–42, 2012), a heavy tail in the degree distribution (Barabási, Albert, Science 286(5439):509–512, 1999) and a high clustering coefficient (Luce, Perry, Psychometrika 14(2):95–116, 1949; Watts, Strogatz, Nature 393(6684):440–442, 1998; Amaral et al., Proc Natl Acad Sci USA 97(21):11149–11152, 2000). In a similar manner, researchers have also been curious about how many polygons are present in a given online social network. Although much effort has been expended (Dantzig et al., International symposium on theory of graphs, p 77–83, 1967; Kamae, IEEE Trans Circuit Theory 14(2):166–171, 1967; Gotlieb, Corneil, Commun ACM 10(12):780–783, 1967; Welch, J ACM 13(2):205–210, 1966; Tiernan, Commun ACM 13(12):722–726, 1970; Tarjan, SIAM J Comput 2(3):211–216, 1973; Johnson, SIAM J Comput 4:77–84, 1975; Mateti, Deo, SIAM J Comput 5(5):90–99, 1976; Marinari et al., Europhys Lett 73(3):301–307, 2005), studying this subject, the inability to enumerate polygons has stymied an in depth understanding of the properties of polygons in an online social network. In the study described in this paper, the estimated number of polygons in an online social network is revealed. It was found that in the current widely used online social networks, e.g., Facebook, Twitter, the number of polygons increases drastically when the length of a polygon is below a set value and then it decreases rapidly. The average length of the network polygons was calculated and it was found that online social networks contain a relatively large average length of polygons. Based on this perspective, a massive labyrinth of polygons would make the online social networks appear to be very complicated. To further investigate this area, a generalized clustering coefficient was explored. Results showed that the generalized clustering coefficient appeared to descend exponentially with the length of polygon and expeditiously approached zero. This result suggested that the polygons with large lengths should be ignored in many scenarios. Since the polygons with lengths greater than five appeared to have little impact on the network, the online social networks appeared to be less complex than anticipated. The polygon is one of the fundamental problems in graph theory and complex networks, so that the work reported here may be beneficial for many disciplines, including transportation [7], engineering [8], computer science [9–16], physics (Birmelé et al., Proceedings of the 24th annual ACM–SIAM symposium on discrete algorithms, p 1884–1896, 2013), sociology (Motter, Albert, Phys Today 65:43, 2012), epidemiology (Feld, Am J Sociol 96(6):1464–1477, 1991; Cohen et al., Phys Rev Lett 91(24):12343, 2002), psychology (Sun et al., Sci Rep 4(6188):5099–5099, 2014), biology (Feiler, Kleinbaum, Psychol Sci, 2015), medicine (Kincaid, Pilette, Comput Appl Biosci Cabios 8:267–273, 1992), geography (Kim et al., Lancet 386(9989):145–153, 2015), etc. [ABSTRACT FROM AUTHOR]

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak and Tuza [SIAM J. Discrete Math., 7 (1994), pp. 307--313] and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon, Krivelevich, and Sudakov [J. Combin. Theory Ser. B, 77 (1999), pp. 73--82] bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for X (G) as a function of the number of vertices n, the number of edges m, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds. [ABSTRACT FROM AUTHOR]

We consider the symmetric difference of two graphs on the same vertex set [n], which is the graph on [n] whose edge set consists of all edges that belong to exactly one of the two graphs. Let F be a class of graphs, and let MF(n) denote the maximum possible cardinality of a family G of graphs on [n] such that the symmetric difference of any two members in G belongs to F. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379-403] with the aim of providing a new graphic approach to coding theory. In particular, MF(n) denotes the maximum possible size of this code. Existing results show that as the graph class F changes, MF(n) can vary from n to 2(1+0(1))(2n). We study several phase transition problems related to MF(n) in general settings and present a partial solution to a recent problem posed by Alon et al. [ABSTRACT FROM AUTHOR]

For integers k ≥1 and m ≥2, let g(k,m) be the least integer n ≥1 such that every graph with chromatic number at least n contains a (k + 1)-connected subgraph with chromatic number at least m. Refining the recent result of Gir\~ao and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868--875] that g(k 1, k) ≤ 7k + 1 for all k≥ 2, we prove that g(k,m) ≤ max(m + 2k 2, (3 + 1/16)k≤) for all k ≥ 1 and m≥2. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367-371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567-586], and of Penev, Thomass'e, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592--619]. Our result implies that g(k, k + 1) (3 + 1/16)k for all k \geq 1, making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261--271] that g(k, k +1)≤3k +1, which was originally a result with a false proof and was the starting point of this research area. [ABSTRACT FROM AUTHOR]

The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer p ≥ 2 , we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power G p of G. This extends some characterizations of graphs with connected medians (case p = 1 ) provided by Bandelt and Chepoi (SIAM J Discrete Math 15(2):268–282, 2002. https://doi.org/10.1137/S089548019936360X). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have G 2 -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop an approach to analyze organizational vulnerability caused by its employees. The proposed approach is adapted from general attack graph analysis approach and social network analysis approach. The attack graph, which is relationship graph in this proposed approach, is created from organization's email logs and virus reports. The relationship graph is analyzed using shortest path analysis to discover all possible attack paths start from risky employees to target employee, and then grouped by path length for further actions based on security policy. The proposed approach was tested using datasets that are limited to only one month with assumption that weight on all edges are equal. This paper suggested further study to improve accuracy of the proposed approach using other mathematical methods such as shortest path analysis with weight or Markov Chains. The proposed approach could also be used by security audit in risk assessment process. [ABSTRACT FROM AUTHOR]

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O (1. 1389 n) for graphs with maximum degree 3 and in time O (1. 2356 n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O (2. 2356 n) as well as an exponential-space O (1. 2330 n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström's compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties. [ABSTRACT FROM AUTHOR]

The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝd, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018]. [ABSTRACT FROM AUTHOR]

Live shrimp movements pose a potential route for site‐to‐site transmission of acute hepatopancreatic necrosis disease (AHPND) and other shrimp diseases. We present the first application of network theory to study shrimp epizootiology, providing quantitative information about the live shrimp movement network of Thailand (LSMN), and supporting practical and policy implementations of disease surveillance and control measures. We examined the LSMN over a 13‐month period from March 2013 to March 2014, with data obtained from the Thailand Department of Fisheries. The LSMN had a mixture of characteristics both limiting and facilitating disease spread. Importantly, the LSMN exhibited power‐law distributions of in and out degrees with exponents of 2.87 and 2.17, respectively. This characteristic indicates that the LSMN behaves like a scale‐free network and suggests that an effective strategy to control disease spread in the Thai shrimp farming sector can be achieved by removing a small number of targeted inter‐site connections (arcs between nodes). Specifically, a disease‐control algorithm based on betweenness centrality (defined as the number of shortest paths between node pairs that traverse a given arc) is proposed here to prioritize targets for disease surveillance and control measures. [ABSTRACT FROM AUTHOR]

Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al. [Maximally recoverable codes for grid-like topologies, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 2092--2108] studied the alphabet size needed for such codes in grid topologies and gave a combinatorial characterization for it. Consider a labeling of the edges of the complete bipartite graph Kn,n, with labels coming from F2d, that satisfies the following condition: for any simple cycle, the sum of the labels over its edges is nonzero. The minimal d where this is possible controls the alphabet size needed for maximally recoverable codes in n×n grid topologies. Prior to the current work, it was known that d is between (log n)² and n log n. We improve both bounds and show that d is linear in n. The upper bound is a recursive construction which beats the random construction. The lower bound follows by first relating the problem to the independence number of the Birkhoff polytope graph, and then providing tight bounds for it using the representation theory of the symmetric group. [ABSTRACT FROM AUTHOR]