Miniconference on the Mathematics of Computation University of Manitoba Combinatorics Seminar Series Burning spiders and path forests Anthony Bonato Ryerson University

Emotions are contagious (Kramer,Guillory,Hancock,14): study of emotional or social contagion in Facebook the underlying network is an essential factor in-person interaction and nonverbal cues are not necessary for the spread of the contagion Graph burning - Anthony Bonato

Miniconference on the Mathematics of Computation Graph burning - Anthony Bonato

Modeling social influence Miniconference on the Mathematics of Computation Modeling social influence general framework: nodes are active or inactive active nodes are introduced and influence the activity of their neighbours Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Models various models: (Kempe, Kleinberg, Tardos,03) competitive diffusion (Alon, et al, 2010) literature in graph theory: domination firefighting percolation... Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Memes memes: an idea, behavior, or style that spreads from person to person within a culture Graph burning - Anthony Bonato

Quantifying meme outbreaks meme breaks out at a node, then spreads to its neighbors over time meme also breaks out at other nodes how long does it take for all nodes to receive the meme in the network? Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Burning number G a connected, simple graph; discrete rounds each node is either burning or non-burning if a node is burning, then it remains in that state every round, choose an additional non-burning node to burn once a node is burning in round t, in round t + 1, each of its non-burning neighbors becomes burning chosen nodes: activators process ends when all nodes are burning the burning number of a graph G, written by b(G), is the minimum number of rounds for all nodes to be burning well-defined, as bounded above by |V(G)| Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Example: cliques b(Kn) = 2 Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Paths 3 2 1 2 3 3 2 3 3 burning sequence: (v3,v7,v9) Theorem (Bonato,Janssen,Roshanbin,14) b(Pn) = 𝑛 . v1 v2 v3 v4 v5 v6 v7 v8 v9 Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Proof of lower bound suppose (x1,…,xk) is a burning sequence for Pn then: Nk-1[x1] ∪ Nk-2[x2] ∪ ⋯ ∪ N0[xk] = V(G) (1) as |Ni(x)| ≤ 2i for all nodes x, we have by (1) that: 2(k−1) + 2(k−2) +⋯+ 2 + k = 2k(k-1)/2 + k = k2 ≥ n Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Trees rooted tree partition of G: collection of rooted trees which are subgraphs of G, with the property that the node sets of the trees partition V(G) x1, x2, x3 are activators Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Trees Theorem (BJR,14) b(G) ≤ k iff there is a rooted tree partition with trees T1,T2,…,Tk of height at most k-1, k-2, …,0 (respectively) such that for all i, j, the roots of Ti and Tj are distance at least |i-j|. Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Trees note: if H is a spanning subgraph of G, then b(G) ≤ b(H) a burning sequence for H is also one for G Corollary (BJR,14) b(G) = min{b(T): T is a spanning tree of G} Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Bounds Corollary (BJR,14) b(Cn) = 𝑛 . If G has a Hamiltonian path, then b(G) ≤ 𝑛 . burning is not monotonic in general on subgraphs; even for isometric subgraphs eg b(C5) = 3 > b(W5) = 2 Lemma (BJR,14) If H is an isometric subtree of G, then b(H) ≤ b(G). Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Aside: spider graphs SP(3,5): Lemma (BJR,14) b(SP(s,r)) = r+1. Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Bounds Theorem (BJR,14) If G has diameter d and radius r, then 𝑑+1 ≤ b(G) ≤ r+1. tight: upper bound: spider graphs lower bound: paths Graph burning - Anthony Bonato

How large can b(G) be? Burning Conjecture (BJR,14) If G has order n, then b(G) ≤ n . Graph burning - Anthony Bonato

Best known upper bounds (Bessy,BJR,Reutanbach,17) b(G) ≤ 1.309 n +3 (Land,Lu,17) b(G) ≤ 1.22 n +2 Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Cartesian grids Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Cartesian grids Theorem (BJR,14) If 1 ≤ m ≤ n, and G is the m x n Cartesian grid, then: If m = O( 𝑛 ), then b(G) = Θ 𝑛 . If m = Ω( 𝑛 ), then b(G) = Θ 𝑚𝑛 1/3 . Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Complexity Burning number problem: Instance: A graph G and an integer k ≥ 2. Question: Is b(G) ≤ k? in NP Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Burning a graph is hard Theorem (BBJRR,17) The Burning number problem is NP-hard. Further, it is NP-hard when restricted to any one of the following graph classes: trees with maximum degree three (!) spiders (!!) path forests (!!!) reduction from partition problems Graph burning - Anthony Bonato

Graph burning - Anthony Bonato How can we burn spiders? arbitrary path lengths and number of branches delete head of spider: a path forest Idea: bounds on burning number of path forests Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Spiders satisfy BC Theorem (B,Lidbetter,17+) Spider graphs satisfy BC: If G has order n, then b(G) ≤ n . Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Two lemmas Lemma 1 (BL,17+) If G is a path forest of order n with t components, then b G ≤ n 2t +t. Lemma 2 (BL,17+) Let G be a family of connected graphs and G’ a subset of G. Suppose that for all G in G’, there is a some vertex v in G and some r≤ V G 1 2 −1 such that either Nr[v] = V(G) or Nr[v] has order at least 2 V G 1 2 −1 and The subgraph induced by V(G) \ Nr[v] is connected and in G. If G satisfies the BC for all G in G \ G’, then G satisfies the BC for all G in G. Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Sketch of Proof Small cases: Lemma 2 and by hand α = n Lemma 2: all arms have length at most 2α -2 Case 1: G has α-1 arms of length α+1 Case 2: Otherwise. Graph burning - Anthony Bonato

Graph burning - Anthony Bonato y = g(t), α = 36 Graph burning - Anthony Bonato

Graph burning - Anthony Bonato y = h(α) - (α-1) Graph burning - Anthony Bonato

Graph burning - Anthony Bonato y = j(α) - (α-1) Graph burning - Anthony Bonato

Graph burning - Anthony Bonato Questions burning conjecture other families of trees? improve general bound? (BL,17+) 3/2-approximation algorithm for computing burning number of path-forests can this be improved? vary rules of burning? vertices burn if joined to r >1 burning vertices Graph burning - Anthony Bonato