Antistrong Digraphs St ephane Bessy University of Montpellier, - PowerPoint PPT Presentation

Antistrong Digraphs St´ ephane Bessy University of Montpellier, LIRMM, France Joint work with: Jœrgen Bang-Jensen (University of South Denmark), Bill Jackson (Queen Mary University of London, UK) and Matthias Kriessel (Universit¨ at Hamburg, Germany) 2015

Antidirected path ◮ In a digraph D , an antidirected path is a path in which the arcs alternate and beginning and ending with a forward arc. x y

Antidirected path ◮ In a digraph D , an antidirected path is a path in which the arcs alternate and beginning and ending with a forward arc. x y ◮ Motivation: find similar (algorithmic) results between directed paths and antidirected paths, but...

Antidirected path ◮ In a digraph D , an antidirected path is a path in which the arcs alternate and beginning and ending with a forward arc. x y ◮ Motivation: find similar (algorithmic) results between directed paths and antidirected paths, but... Theorem (A. Yeo, 2014) Given two vertices x and y of D, it is NP-complete to decide if D admits an antidirected path from x to y.

Antidirected trail ◮ An antidirected trail is a trail (no repeated arc) in which the arcs alternate and beginning and ending with a forward arc. x y

Antidirected trail ◮ An antidirected trail is a trail (no repeated arc) in which the arcs alternate and beginning and ending with a forward arc. Theorem It is polynomial to check if there exists an antidirected trail from x to y. Proof : B ( D ): the (oriented) adjacency bipartite representation of D . − − − − 1 2 3 4 1 3 4 2 + + + + 1 2 3 4 D B(D)

Antistrong digraph ◮ A digraph is antistrong if for x , y ∈ V ( D ) there exists an andirected trail from x to y . Theorem For | D | ≥ 3 , D is antistrong iff B ( D ) is connected.

Antistrong digraph ◮ A digraph is antistrong if for x , y ∈ V ( D ) there exists an andirected trail from x to y . Theorem For | D | ≥ 3 , D is antistrong iff B ( D ) is connected. ◮ We provide some algorithmic results related to ’antistrongness’.

Antistrong digraph ◮ A digraph is antistrong if for x , y ∈ V ( D ) there exists an andirected trail from x to y . Theorem For | D | ≥ 3 , D is antistrong iff B ( D ) is connected. ◮ We provide some algorithmic results related to ’antistrongness’. ◮ First easy one: in polytime we can check ’antistrongness’.

Direct results: k -antistrong digraph ◮ D is k -antistrong if for every x , y ∈ D there exist k -arc-disjoint antidirected trails from x to y . Theorem D is k-antistrong iff B ( D ) is k-edge-connected.

Direct results: k -antistrong digraph ◮ D is k -antistrong if for every x , y ∈ D there exist k -arc-disjoint antidirected trails from x to y . Theorem D is k-antistrong iff B ( D ) is k-edge-connected. Corollaries: ◮ In polytime we can check ’ k -antistrongness’.

Direct results: k -antistrong digraph ◮ D is k -antistrong if for every x , y ∈ D there exist k -arc-disjoint antidirected trails from x to y . Theorem D is k-antistrong iff B ( D ) is k-edge-connected. Corollaries: ◮ In polytime we can check ’ k -antistrongness’. ◮ If D is 2 k -antistrong then D contains k arc-disjoint spanning antistrong subdigraphs.

Direct results: a matro¨ ıd for antistrongness ◮ A CAT or close antidirected trail is an alternating close trail. ◮ The cat-free sets of arcs of D form a matro¨ ıd on the arcs of D .

Our main results: orientations

Our main results: orientations CAT-free orientation: Theorem Let G = ( V , E ) with | E | ≤ 2 | V | − 1 . G has a CAT-free orientation iff: | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2)

Our main results: orientations CAT-free orientation: Theorem Let G = ( V , E ) with | E | ≤ 2 | V | − 1 . G has a CAT-free orientation iff: | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) Remarks: ◮ (1) and (2) are necessary. ◮ No bipartite digraph is antistrong.

Cat-free orientation Proof: In two steps: ◮ A graph is an odd-pseudoforest if each of its connected component contains a most one cycle which is odd if it exists.

Cat-free orientation Proof: In two steps: ◮ A graph is an odd-pseudoforest if each of its connected component contains a most one cycle which is odd if it exists. ◮ Claim 1: G satisfies (1) and (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest.

Cat-free orientation Proof: In two steps: ◮ A graph is an odd-pseudoforest if each of its connected component contains a most one cycle which is odd if it exists. ◮ Claim 1: G satisfies (1) and (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation.

Cat-free orientation ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation. Proof: r Y � � � � Tree X Odd pseudoforest

Cat-free orientation ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation. Proof: r Y � � � � Tree X Odd pseudoforest

Cat-free orientation ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation. Proof: r Y � � � � Tree X Odd pseudoforest

Cat-free orientation ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation. Proof: r Y � � � � Tree X Odd pseudoforest

Cat-free orientation ◮ Claim 2: Every graph which is the (edge)-union of a forest and an odd pseudoforest admits a cat-free orientation. Proof: r Y � � � � Tree X Odd pseudoforest

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: graph theory vs matroids

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: graph theory vs matroids ◮ Let E be a set and f : 2 E → Z a submodular, nondecreasing function which is nonnegative on 2 E \ {∅} . Theorem (J. Edmonds, 1970) The sets I ⊆ E s.t. ∀∅ � = I ′ ⊆ I | I ′ | ≤ f ( I ′ ) form a matroid M f on E. The rank of a subset S ⊆ E in M f is given by the min-max formula: � � | S 0 | + � k r f ( S ) = min ( S 0 , S 1 ,..., S k ) i =1 f ( S i ) where the min is taken over all partitions ( S 0 , S 1 , . . . , S k ) of S.

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: ◮ Ex1 : f = ν − 1, where ν ( I ) is the nb of vertices incident with edges in I , the cycle matroid .

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: ◮ Ex1 : f = ν − 1, where ν ( I ) is the nb of vertices incident with edges in I , the cycle matroid . ◮ Ex2 : f = ν − β , where β ( I ) is the nb of bipartite components formed by the edges of I , the even bicircular matroid .

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: ◮ Ex1 : f = ν − 1, where ν ( I ) is the nb of vertices incident with edges in I , the cycle matroid . ◮ Ex2 : f = ν − β , where β ( I ) is the nb of bipartite components formed by the edges of I , the even bicircular matroid . ◮ Ex3 : f = 2 ν − 1 − β . Independent in M f iff satisfies (1) and (2).

Cat-free orientation ◮ Claim 1: G satisfies | E ( H ) | ≤ 2 | V ( H ) | − 1 for all ( � = ∅ ) subgraphs H of G (1) | E ( H ) | ≤ 2 | V ( H ) | − 2 for all ( � = ∅ ) bip. subgraphs H of G (2) iff it can be (edge)-partionned into a forest and an odd pseudoforest. Proof: ◮ Every independent of M f ∨ M g is an independent of M f + g , but in general the converse is not true.