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

2. 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

3. Miniconference on the Mathematics of Computation
Graph burning - Anthony Bonato

4. 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

5. 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

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

7. 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

8. 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

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

10. 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 .
v v2 v v v5 v6 v v8 v9
Graph burning - Anthony Bonato

11. 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

12. 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

13. 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

14. 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

15. 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

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

17. 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

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

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

20. Graph burning - Anthony Bonato
Cartesian grids
Graph burning - Anthony Bonato

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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 such that either Nr[v] = V(G) or
Nr[v] has order at least 2 V G −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

27. 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

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

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

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

31. 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