Post on 01-Apr-2015
Towards the Construction of a Fast
Algorithm for the Vertex Separation
Problem on Cactus Graphs
Minko Markov
Sofia University, Faculty of Mathematics and Informatics
minkom@fmi.uni-sofia.bg
Structure of the presentation Background Vertex Separation of Trees and
Unicyclics Vertex Separation of Cacti Boudaried Cacti and Stretchability Decomposition of Boundaried Cacti Main Theorem for Stretchability on
Boundaried Cacti
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Vertex Separation (VS) of Layouts and Graphs An NP-complete problem on undirected
ordinary graphs Do not confuse “Vertex Separation” with
“Vertex Separator” The definition of Vertex Separation is
based on the definition of linear layout
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
vs(G) = min {vsL(G) | L is a layout of G} = 2
VS of Layouts and Graphs (2)
April 11, 2023
vsL(G)=2
0x
y
vu
w
u v w y x
1 2 2 2
L
G = (V,E)
πL(u) = {u}πL(v) = {u,v}πL(w) = {v,w}πL(y) = {w,y}πL(x) = ∅Minko Markov, Faculty of Mathematics and Informatics, Sofia University
(u,v), (u,w), (v,w), (v,y), and (w,y) are clean
Node Search Number (SN)
April 11, 2023Minko Markov, Faculty of Mathematics and Informatics. This research is supported
by Sofia University Science Fund under project "Discrete Structures"
w+
all edges are contaminated(u,v), (u,w) and (v,w) are clean
xy
vu
wu+ v+ x+y+ v—u— y— w—x—
sns(G) = 3
S =
(u,v) is cleanall edges are clean
monotonous (progressive) search
VS is equivalent to SN
For every graph G, vs(G) = sn(G) − 1 Optimal searches define unique optimal
layouts, optimal layouts define multitudes of optimal searches
April 11, 2023
x y
vu
w L = u v w y x, vsL(G) = 2
S = u+ v+ w+ u− y+ v− x+ y− x− w−, sns(G) = 3
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Fast algorithms for VS on restric-ted graphs
O(n) for trees (Ellis, Sudborough, Turner, 1994) O(n lg n) on unicyclic graphs (Ellis, Markov,
2004), improved to O(n) (Chou, Ko, Ho, Chen, 2006)
O(bc + c2 + n) on block graphs (Chou et al., 2008) O(n) on 3-Cycle-Disjoint Graphs—a strict
subclass of cactus graphs (Yang, Zhang, Cao 2010)
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Cactus graphs (cacti)
April 11, 2023
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Rooted Cacti
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS of trees – O(n) algorithm by Ellis, Sudborough, Turner (1994) Theorem (EST, 1994): If T is a tree and
k ≥ 1, then vs(T) ≤ k iff every vertex induces at most two subtrees of vs = k.
April 11, 2023
vvs = k vs = k
< k < k < k
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
k-critical subtree
T is a rooted tree, vs(T) = k, and the root induces two subtrees of vs = k.
April 11, 2023
k k< k < k...
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Label of a tree
April 11, 2023
T2TT1
lab(T) = (k, p, q), k > p > q
p pk k
qvs(T)=kvs(T1)=pvs(T2)=q
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The EST algorithm
April 11, 2023
▲ _ _ ▲ _ _ ▲ _ lab:
lab = ?
lab1 = (5,2)
lab1:9 8 7 6 5 4 3 2 1
lab2 = (7,6,5)
lab3 = (8,5,2c)
lab2:
lab3:
_ _ _ ▲ _ _ ● _ _ ▲ ▲ ● _ _ _ _
● _ _ _ _ _ _ _ _ ● _ _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ _ _ _ _ ● _ _ ● _ _ ● _ _ _
lab = (9)
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The VS backbone of a tree the easiest kind of rooted tree of VS k
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS < k VS < kVS < k
VS = k1
K−1 K−1 K−1
The VS backbone of a tree the second best kind (VS = k)
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS = kVS < kVS < k
VS = k1
K K−1 K−1
The VS backbone of a tree an even harder rooted tree of VS k
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS = k VS = kVS < k
VS = k1
K K−1 K
The VS backbone of a tree the hardest kind
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS = k VS = kVS < k
VS = k
1
K K−1 K
VS = k-1 VS = k-1
1
1
K−1 K−1
The backbone of a non-rooted tree
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
VS < k
VS = k1
K−1 K−1 K−1K−1 K−1 K−1 K−1 K−1 K−1
1 1
Vertex Separation of Cacti
Theorem (M.M., 2007). Let G be a cactus and k ≥ 1. Then vs(G) ≤ k iff:Every vertex induces at most two cacti of
separation k, all others are < k.In every cycle there exist vertices u and v
(not necessarily distinct) such that G [u,v]⊝ is k-stretchable.
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
G⊝[u,v]
April 11, 2023
u v
G G [u,v]⊝
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Stretchability k w.r.t. u and v
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
u v
K
The idea behind the theorem Definition: a c-path (cactus path) in a cactus
is a linear order of vertices and cycles
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
a
b
e
g
c
kid
tq
rponm
lj
hf vu xs3
s2s1
w
s4
C = a s1 f g h s2 m n o s3 u v w x
The backbone of a cactus
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
bfd
he
ca is
j
K−1 K−1 K−1 K−1 K−1 K−1K−1K−1
K
The root and the backbone
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
r
bfd
he
ca is
jK
lab(G(r)) = ( K,
G
lab(G1(r)) )
G1
The root and the backbone
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
r
fd
he
s
lab(G(r)) = ( K,
i jb caK
G
lab(G1(r)) )
G1
ic
The cacti pitfall
April 11, 2023
k-1 k-1
k k
k k
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The cacti pitfall
April 11, 2023
k k
k-2 k-2
k k
Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The cacti pitfall
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The solution for cacti?
Take Stretchability w.r.t. k vertex pairs as the primary problem
Consider bounaried cacti, the boundary being the vertices w.r.t. which we stretch
The original problem reduces to this one – just take an empty boundary
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
Boundaried cactus
A cactus G in which some cycles s1, …, sn have two boundary vertices each. All boundary vertices are of degree 2.
Let the boundary pair in si be ‹ui, wi›. The search game on G is performed so that n searchers are placed on U = {u1, …, un} initially and at the end, each of W = {w1, …, wn} must have a searcher.
The boundary is ‹U, W›.
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
The Residual VS of a boundaried cactus Let G be a boundaried cactus with n
vertex pairs in the boundary. Let k be the stretchability of G w.r.t. the boundary. Then rvs(G) = k – n. We proved k – n > 0 always.
From now on we consider RVS of boundaried cacti. VS of cacti is a special case of RVS.
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
RVS of Boundaried Cacti
Theorem: Let G be a boundaried cactus, boundary ‹U, W›, and m ≥ 1. Then rvs(G) ≤ m iff:Every nonboundary vertex induces at most two
boundaried cacti of rvs m, all others are < m.In every cycle there are nonboundary vertices x
and y (not necessarily distinct) such that G is (k+1)-stretchable w.r.t. ‹U {x}, W {y}› or ‹U {y}, W {x}›.
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
How to prove k-stretchability It is more rigorous to use the VS
definition and terminology, not the NSN
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
L is k-stretchable iff the separation of any vertex is (k – the number of intervals it is in)
u rx
{u,v,w} : left, {x,y} : right
v wy
r is the rightmost neighbour of x and y
layout L
How to prove k-stretchability It is easier to modify L into an extended
layout L* and consider its VS
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
xu v wy
layout Llayout L*
u v w
L is k-stretchable iff L* has VS ≤ k
Proof of the theorem, part I Consider an optimal extended layout L*.
Consider the leftmost and rightmost nonboundary vertices a and z.
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University
az
s3s2s1 s4
rvs(G) ≤ 5 → rvs(G1) ≤ 4, i.e. vs(G1) ≤ 4
G1
G
THE END
April 11, 2023 Minko Markov, Faculty of Mathematics and Informatics, Sofia University