Bicoloured Graphs Dually Connectedness Dual Separators and Beyond CAI Leizhen The Chinese Univ of Hong Kong Joint work with YE Junjie 2 3 4 Corneilian Graph Charles Mark Lorna ID: 475706
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Edge Bicoloured Graphs: Dually Connectedness, Dual Separators, and Beyond
CAI
Leizhen
The Chinese
Univ
of Hong Kong
Joint work with YE
JunjieSlide2
2Slide3
3Slide4
4
Corneilian
Graph
Charles
Mark
Lorna
Leizhen
Gara
Jim
JasonSlide5
5
Definitions
Edge
bicoloured
graph G: every edge is
coloured
either
blue
or
red
.
Dually connected: both
blue graph
and
red graph
are connected.
Dual separator: vertices whose deletion disconnects both
and
.
Remark: Definitions of bicoloured graphs can differ by allowing multiedges or multicolours, and definitions of blue (red) graphs can differ by isolated vertices.
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6
Problems
Dully Connected Induced Subgraph
Find k vertices V’ to induce a dually connected subgraph.
Dully Connected Deletion
Delete k vertices V’ to obtain a dually connected graph.
Remark: DCIS(k) = DCD(n-k)
Dual Separator
Delete fewest vertices to disconnect both blue and red graphs.Slide7
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Largest dually connected subgraphDoes G have at least k vertices V’ to make G[V’] dually connected?
Straightforward in O(mn) time.Slide8
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Dually connected induced subgraphDoes G have k vertices V’ such that G[V’] is dually connected?
Theorem 1. Cubic time
for
bicoloured
complete
graphs,
but NP-complete and W[1]-hard for dual trees
(and
tricoloured
complete graphs).
Dual tree: both blue and red graphs are spanning trees.
Theorem 2. NP- and W[1]-hard to find a strongly connected induced
subdigraph
on k vertices.Slide9
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Dually connected subgraph: NPC
Reduction from k-CliqueSlide10
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Dually connected subgraph: NPC
Reduction from k-CliqueSlide11
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Dually connected subgraph: NPC
Reduction from k-CliqueSlide12
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Dually connected subgraph: NPC
Reduction from k-CliqueSlide13
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Dually connected subgraph: NPC
Reduction from k-CliqueSlide14
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Dually connected: k-vertex deletionDoes G contain k vertices V’ such that G – V’ is dually connected?
Theorem 3. NP-complete and W[1]-hard, but FPT for dual trees.
Theorem 4. NP- and W[1]-hard to delete k vertices to form a strongly connected digraph.Slide15
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Dually connected: k-vertex deletion
Reduction from Independent k-Set.
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Dually connected: k-vertex deletion
Reduction from Independent k-Set.
Slide17
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Dually connected: k-vertex deletion
Reduction from Independent k-Set.
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FPT algorithm for dual trees T
Lemma 5. T has a solution
iff
it has k vertices covering 2k edges.Slide19
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FPT algorithm for dual trees
Minor modification of a random separation algorithm of
Cai
,
Chan
and Chan for finding vertices to cover exactly k edges.
Main ideas:
use a random vertex-partition to separate a solution
into
components,
find appropriate components to form a solution, and
derandomize
the algorithm by (n,3k)-universal sets.
Time:
Slide20
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FPT algorithm for dual treesSlide21
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Dual separatorDoes G contain at most k vertices whose deletion disconnects both blue and red graphs?
Dual (
s,t
)-Separator:
Does G contain at most k vertices whose deletion disconnects (
s,t
)-paths in both blue and red graphs?
Theorem 6. NP-complete for both Dual (
s,t
)-Separator and Dual Separator problems.
Theorem 7. NP-complete to delete k vertices to disconnect both (
s,t
)- and (
t,s
)-paths in a digraph.Slide22
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Dual separator: NP-completeness
Idea for Dual (
s,t
)-Separator:
R
eduction from Vertex Cover on bipartite graphs. Slide23
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Dual (s,t)-separator: NP-completeness
Reduction from Vertex Cover on cubic graphs.
Edge partition a cubic graph into two spanning bipartite graphs.Slide24
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Conclusion: generalization to hereditary (
,
)-subgraph
Does G have k vertices V’ such that
and
, respectively, are
- and
-graphs?
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Conclusion: general vertex deletion
Theorem 8. Let
be graphs, and
graph properties
characterizable
by finite forbidden induced
subgraphs
.
It is FPT to determine whether there are k vertices
S
in
such that
each
S
is
a
-
graph
.
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Open problems
Find largest dually connected induced subgraph in time
?
Determine parameterized complexity of Dual (
s,t
)-Separator and Dual Separator. Any connection with network flows?
Parameterized complexity of Dual
-Graph Deletion for
being acyclic, bipartite,
chordal
, and planar graphs, respectively.
Complexity of turning edge
bicoloured
graphs into dual
-graphs for
being acyclic, bipartite,
chordal
, and planar graphs, respectively.
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