110 research outputs found

    Investigations in the semi-strong product of graphs and bootstrap percolation

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    The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g1,h1)(g2,h2) is an edge of the product whenever g1g2 is an edge of G and h1h2 is an edge of H or g1 = g2 and h1h2 is an edge of H. A natural subject for investigation is to determine properties of the semi-strong product in terms of those properties of its factors. We investigate distance, independence, matching, and domination in the semi-strong product Bootstrap Percolation is a process defined on a graph. We begin with an initial set of infected vertices. In each subsequent round, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected, vertices remain infected. The parameter r is called the percolation threshold. When G is finite, the infection either stops at a proper subset of G or all of V(G) becomes infected. If all of V(G) eventually becomes infected, then we say that the infection percolates and we call the initial set of infected vertices a percolating set. The cardinality of a minimum percolating set of G with percolation threshold r is denoted m(G,r). We determine m(G,r) for certain Kneser graphs and bipartite Kneser graphs

    Minimum Target Sets in Non-Progressive Threshold Models: When Timing Matters

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    Let GG be a graph, which represents a social network, and suppose each node vv has a threshold value Ï„(v)\tau(v). Consider an initial configuration, where each node is either positive or negative. In each discrete time step, a node vv becomes/remains positive if at least Ï„(v)\tau(v) of its neighbors are positive and negative otherwise. A node set S\mathcal{S} is a Target Set (TS) whenever the following holds: if S\mathcal{S} is fully positive initially, all nodes in the graph become positive eventually. We focus on a generalization of TS, called Timed TS (TTS), where it is permitted to assign a positive state to a node at any step of the process, rather than just at the beginning. We provide graph structures for which the minimum TTS is significantly smaller than the minimum TS, indicating that timing is an essential aspect of successful target selection strategies. Furthermore, we prove tight bounds on the minimum size of a TTS in terms of the number of nodes and maximum degree when the thresholds are assigned based on the majority rule. We show that the problem of determining the minimum size of a TTS is NP-hard and provide an Integer Linear Programming formulation and a greedy algorithm. We evaluate the performance of our algorithm by conducting experiments on various synthetic and real-world networks. We also present a linear-time exact algorithm for trees.Comment: Accepted in ECAI-23 (26th European Conference on Artificial Intelligence

    Virginia Dental Journal (Vol. 99, no. 2, 2022)

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    Local treewidth of random and noisy graphs with applications to stopping contagion in networks

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    We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all kk-vertex subgraphs of an nn-vertex graph. When kk is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is kk, the number of "infected" vertices in the network. For a certain range of parameters the running time of our algorithms on nn-vertex graphs is 2o(k)poly(n)2^{o(k)}\textrm{poly}(n), improving upon the 2Ω(k)poly(n)2^{\Omega(k)}\textrm{poly}(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems

    Exploring the barriers to compassion for postpartum mothers and their experiences during the COVID-19 pandemic

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    This portfolio thesis comprises of three parts:Part One: Systematic Literature ReviewThe systematic literature review explored the psychological experiences of postpartum mothers during the COVID-19 pandemic. A systematic search of 5 electronic databases found twelve qualitative papers that met the inclusion criteria. The National Institute for Health and Care Excellence (NICE) Quality Appraisal Checklist for Qualitative Studies was used to evaluate the quality of the studies, whilst Thomas and Harden’s (2008) Thematic Synthesis was used to configure the findings across the studies. Four superordinate themes were identified: relationships, psychological strengths, mental health difficulties, and emotional responses. The findings revealed a range of positive and negative psychological experiences, with some postpartum mothers experiencing psychological growth, and others emotional distress. Clinical implications and key areas for future research are discussed.Part Two: Empirical PaperThe empirical paper explored the fears, blocks and resistances (FBRs) to the flows of compassion in first-time mothers. Nine women attended an online semi-structured interview with the researcher that were analysed using Reflexive Thematic Analysis (Braun & Clark, 2019). The study found three themes, with subthemes, that encompassed the FBRs that mothers experienced: ‘Super Mum’: the Unobtainable Ideal, the Exchange of Distress and Compassion, and Going Through it Alone. The FBRs identified within each theme are summarised, and a theme map illustrates the relationships between themes and how this maintains FBRs for first-time mothers. Clinical implications of the research and areas for future research are discussed.Part Three: AppendicesAppendices relevant to the systematic literature review and empirical paper, including a reflective statement, epistemological statement, and all relevant documentation

    Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks

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    We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all k-vertex subgraphs of an n-vertex graph. When k is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive nearly tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is k, the number of initially "infected" vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on n-vertex graphs is 2^o(k) poly(n), improving upon the 2^?(k) poly(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems

    In Memoriam, Solomon Marcus

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    This book commemorates Solomon Marcus’s fifth death anniversary with a selection of articles in mathematics, theoretical computer science, and physics written by authors who work in Marcus’s research fields, some of whom have been influenced by his results and/or have collaborated with him

    Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

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    Let r≥1\mathcal{r} ≥ 1 be any non negative integer and let G=(V,E)G = (V, E) be any undirected graph in which a subset D⊆VD ⊆ V of vertices are initially infected. We consider the following process in which, at every step, each non-infected vertex with at least r\mathcal{r} infected neighbours becomes and an infected vertex never becomes non-infected. The problem consists in determining the minimum size sr(G)s_r (G) of an initially infected vertices set DD that eventually infects the whole graph GG. Note that s1(G)s_1 (G) = 1 for any connected graph GG. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that s1(G)=1s_1(G) = 1 for any connected graph GG. The case when GG is the n×nn × n grid Gn×nG_{n×n} and r=2\mathcal{r} = 2 is well known and appears in many puzzles books, in particular due to the elegant proof that shows that s2(Gn×n)s_2(G_{n×n}) = nn for all nn ∈ N\mathbb{N}. We study the cases of square grids Gn×nG_{n×n} and tori Tn×nT_{n×n} when r\mathcal{r} ∈ {3, 4}. We show that s3(Gn×n)s_3(G_{n×n}) = ⌈n2+2n+43⌉\lceil\frac{n^2+2n+4}{3}\rceil for every nn even and that ⌈n2+2n3⌉\lceil\frac{n^2 +2n}{3}\rceil ≤ s3(Gn×n)s_3(G_ {n×n}) ≤ ⌈n2+2n3⌉\lceil\frac{n^2 +2n}{3}\rceil + 1 for any nn odd. When nn is odd, we show that both bounds are reached, namely s3(Gn×n)s_3(G_{n×n}) = ⌈n2+2n3⌉\lceil\frac{n^2 +2n}{3}\rceil if nn ≡ 5 (mod 6) or nn = 2p^p − 1 for any pp ∈ N∗\mathbb{N}^*, and s3(Gn×n)s_3(G_{n×n}) = ⌈n2+2n3⌉\lceil\frac{n^2 +2n}{3}\rceil + 1 if nn ∈ {9, 13}. Finally, for all nn ∈ N\mathbb{N}, we give the exact expression of s4(Gn×n)s_4(G_{n×n}) and of sr(Tn×n)s_r(T_{n×n}) when r\mathcal{r} ∈ {3, 4}
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