Finding good 2-partitions of digraphs I. Hereditary properties

Theoretical Computer Science

2019

A (δ ≥ k1, δ ≥ k2)-partition of a graph G is a vertex-partition (V1, V2) of G satisfying that δ(G[Vi]) ≥ ki for i = 1, 2. We determine, for all positive integers k1, k2, the complexity of deciding whether a given graph has a (δ ≥ k1, δ ≥ k2)-partition. We also address the problem of finding a function g(k1, k2) such that the (δ ≥ k1, δ ≥ k2)-partition problem is N P-complete for the class of graphs of minimum degree less than g(k1, k2) and polynomial for all graphs with minimum degree at least g(k1, k2). We prove that g(1, k) = k for k ≥ 3, that g(2, 2) = 3 and that g(2, 3), if it exists, has value 4 or 5.A 2-partition of a graph G is a partition of V (G) into two disjoint sets. Let P 1 , P 2 be two graph properties, then a (P 1 , P 2 )-partition of a graph G is a 2-partition (V 1 , V 2 ) where V 1 induces a graph with property P 1 and V 2 a graph with property P 2 . For example aThere are many papers dealing with vertex-partition problems on (di)graphs. Examples (from a long list) are Examples of 2-partition problems are recognizing bipartite graphs (those having has a 2-partition into two independent sets) and split graphs (those having a 2-partition into a clique and an independent set) [15]. It is well known and easy to show that there are linear algorithms for deciding whether a graph is bipartite, respectively, a split graph. It is an easy exercise to show that every graph G has a 2-partition (V 1 , V 2 ) such that the degree of each vertex in G[V i ], i ∈ [2] is at most half of its original degree. Furthermore such a partition can be found efficiently by a greedy algorithm. In [16,17] and several other papers the opposite condition for a 2-partition was studied. Here we require the that each vertex has at least half of its neighbours inside the set it belongs to in the partition. This problem, known as the satisfactory partition problem, is N P-complete for general graphs [5].A partition problem that has received particular attention is that of finding sufficient conditions for a graph to possess a (δ ≥ k 1 , δ ≥ k 2 )-partition. Thomassen [31] proved the existence of a function f (k 1 , k 2 ) so that every graph of minimum degree at least f (k 1 , k 2 ) has a (δ ≥ k 1 , δ ≥ k 2 )-partition.He proved that f (k 1 , k 2 ) ≤ 12 · max{k 1 , k 2 }. This was later improved by Hajnal [19] and Häggkvist (see [31]). Thomassen [31,32] asked whether it would hold that f (k 1 , k 2 ) = k 1 + k 2 + 1 which would be best possible because of the complete graph K k1+k2+1 . Stiebitz [29] proved that indeed we have f (k 1 , k 2 ) = k 1 + k 2 + 1. Since this result was published, several groups of researchers have tried to find extra conditions on the graph that would allow for a smaller minimum degree requirement. Among others the following results were obtained.Theorem 1.1 [20] For all integers k 1 , k 2 ≥ 1 every triangle-free graph G with δ(G) ≥ k 1 + k 2 has a (δ ≥ k 1 , δ ≥ k 2 )-partition.

“…The following observation which forms the base of many of our proofs is easy to prove (for a proof of result a very similar to this see [4]).…”

Section: Ring Graphs and 3-satmentioning

confidence: 69%

Section: Ring Graphs and 3-satmentioning

confidence: 99%

Degree-constrained 2-partitions of graphs

Theoretical Computer Science

2019

“…, Vp) of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by Vi, (1 ≤ i ≤ p) is at least k smaller than the maximum out-degree of D. We show that this problem is polynomial-time solvable when p ≥ 2k and N P-complete otherwise. The result for k = 1 and p = 2 answers a question posed in [3]. We also determine, for all fixed non-negative integers k1, k2, p, the complexity of deciding whether a given digraph of maximum out-degree p has a 2-partition (V1, V2) such that the digraph induced by Vi has maximum out-degree at most ki for i ∈ [2].…”

mentioning

confidence: 81%

Out-degree reducing partitions of digraphs

Theoretical Computer Science

Havet

et al. 2018

Self Cite

Let k be a fixed integer. We determine the complexity of finding a p-partition (V1, . . . , Vp) of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by Vi, (1 ≤ i ≤ p) is at least k smaller than the maximum out-degree of D. We show that this problem is polynomial-time solvable when p ≥ 2k and N P-complete otherwise. The result for k = 1 and p = 2 answers a question posed in [3]. We also determine, for all fixed non-negative integers k1, k2, p, the complexity of deciding whether a given digraph of maximum out-degree p has a 2-partition (V1, V2) such that the digraph induced by Vi has maximum out-degree at most ki for i ∈ [2]. It follows from this characterization that the problem of deciding whether a digraph has a 2-partition (V1, V2) such that each vertex v ∈ Vi has at least as many neighbours in the set V3−i as in Vi, for i = 1, 2 is N P-complete. This solves a problem from [6] on majority colourings.

“…Many other

2

‐partition problems have already been studied. The papers determined the complexity of a large number of

2

‐partition problems where we seek a

2

‐partition

(V_{1}, V_{2})

with specified properties for the digraphs

D ⟨ V_{i} ⟩

induced by this partition. In , Bang‐Jensen and Havet asked whether there exists a polynomial‐time algorithm to decide whether a given digraph has a

2

‐partition

(V_{1}, V_{2})

with

{normalΔ}^{+} (D 〈 V_{i} 〉) < {normalΔ}^{+} (D)

for

i \in {1, 2}

.…”

Section: Introductionmentioning

confidence: 99%

“…The papers determined the complexity of a large number of

2

‐partition problems where we seek a

2

‐partition

(V_{1}, V_{2})

with specified properties for the digraphs

D ⟨ V_{i} ⟩

induced by this partition. In , Bang‐Jensen and Havet asked whether there exists a polynomial‐time algorithm to decide whether a given digraph has a

2

‐partition

(V_{1}, V_{2})

with

{normalΔ}^{+} (D 〈 V_{i} 〉) < {normalΔ}^{+} (D)

for

i \in {1, 2}

. This was answered affirmatively in where also the complexity of deciding whether a digraph

D

has a 2‐partition

(V_{1}, V_{2})

so that

{normalΔ}^{+} (D 〈 V_{i} 〉) \leq k_{i}

was determined for all nonnegative integers

k_{1}, k_{2}

.…”

Section: Introductionmentioning

confidence: 99%

Bipartite spanning sub(di)graphs induced by 2‐partitions

Journal of Graph Theory

Havet

et al. 2018

Self Cite

For a given 2‐partition ( V 1 , V 2 ) of the vertices of a (di)graph G, we study properties of the spanning bipartite subdigraph B G ( V 1 , V 2 ) of G induced by those arcs/edges that have one end in each V i , i ∈ { 1 , 2 }. We determine, for all pairs of nonnegative integers k 1 , k 2, the complexity of deciding whether G has a 2‐partition ( V 1 , V 2 ) such that each vertex in V i (for i ∈ { 1 , 2 }) has at least k i (out‐)neighbours in V 3 − i. We prove that it is scriptNP‐complete to decide whether a digraph D has a 2‐partition ( V 1 , V 2 ) such that each vertex in V 1 has an out‐neighbour in V 2 and each vertex in V 2 has an in‐neighbour in V 1. The problem becomes polynomially solvable if we require D to be strongly connected. We give a characterisation of the structure of scriptNP‐complete instances in terms of their strong component digraph. When we want higher in‐degree or out‐degree to/from the other set, the problem becomes scriptNP‐complete even for strong digraphs. A further result is that it is scriptNP‐complete to decide whether a given digraph D has a 2‐partition ( V 1 , V 2 ) such that B D ( V 1 , V 2 ) is strongly connected. This holds even if we require the input to be a highly connected eulerian digraph.