An Ore-type condition for arbitrarily vertex decomposable graphs

A graph G of order n is arbitrarily partitionable (AP) if, for every sequence (n 1 ,. .. , n p) partitioning n, there is a partition (V 1 ,. .. , V p) of V (G) such that G[V i ] is a connected n i-graph for i = 1,. .. , p. The property of being AP is related to other well-known graph notions, such as perfect matchings and Hamiltonian cycles, with which it shares several properties. This work is dedicated to studying two aspects behind AP graphs. On the one hand, we consider algorithmic aspects of AP graphs, which received some attention in previous works. We first establish the NP-hardness of the problem of partitioning a graph into connected subgraphs following a given sequence, for various new graph classes of interest. We then prove that the problem of deciding whether a graph is AP is in NP for several classes of graphs, confirming a conjecture of Barth and Fournier for these. On the other hand, we consider the weakening to APness of sufficient conditions for Hamiltonicity. While previous works have suggested that such conditions can sometimes indeed be weakened, we here point out cases where this is not true. This is done by considering conditions for Hamiltonicity involving squares of graphs, and claw-and net-free graphs.

Section: Hamiltonian Aspects Of Ap Graphsmentioning

confidence: 99%

More aspects of arbitrarily partitionable graphs

Bensmail¹,

Li²

2020

“…The general idea is that one can consider any of the numerous sufficient conditions for a graph to be Hamiltonian or traceable, and investigate whether it can be weakened to a sufficient condition for APness. This line of research was initiated by Marczyk in [13], in which he focused on the parameter σ 2 , being defined as σ 2 (G) = min {d(u) + d(v) : u and v are independent vertices of G} for any graph G. By a famous result of Ore [16], recall indeed that any connected n-graph G is Hamiltonian whenever σ 2 (G) ≥ n, while G is traceable whenever σ 2 (G) ≥ n − 1. In [13], Marczyk proved that G is AP provided σ 2 (G) ≥ n − 2 and α(G) ≤ ⌈n/2⌉ (that is, provided G has a perfect matching or a quasi-perfect matching).…”

Section: Introductionmentioning

confidence: 99%

“…This line of research was initiated by Marczyk in [13], in which he focused on the parameter σ 2 , being defined as σ 2 (G) = min {d(u) + d(v) : u and v are independent vertices of G} for any graph G. By a famous result of Ore [16], recall indeed that any connected n-graph G is Hamiltonian whenever σ 2 (G) ≥ n, while G is traceable whenever σ 2 (G) ≥ n − 1. In [13], Marczyk proved that G is AP provided σ 2 (G) ≥ n − 2 and α(G) ≤ ⌈n/2⌉ (that is, provided G has a perfect matching or a quasi-perfect matching). Later on, in [9,14], Marczyk, together with Horňák, Schiermeyer, and Woźniak, improved this sufficient condition to G satisfying only σ 2 (G) ≥ n−5 and additional conditions (such as the previous condition on α(G), and also conditions on n).…”

Section: Introductionmentioning

confidence: 99%

A \sigma₃ condition for arbitrarily partitionable graphs

Bensmail¹

2024

A graph G of order n is arbitrarily partitionable (AP for short) if, for every partition (λ 1 , . . . , λ p ) of n, there is a partition (is a connected graph of order λ i for every i ∈ {1, . . . , p}. Several aspects of AP graphs have been investigated to date, including their connection to Hamiltonian graphs and traceable graphs. Every traceable graph (and, thus, Hamiltonian graph) is indeed known to be AP, and a line of research on AP graphs is thus about weakening, to APness, known sufficient conditions for graphs to be Hamiltonian or traceable.In this work, we provide a sufficient condition for APness involving the parameter σ 3 , where, for a given graph G, the parameter σ 3 (G) is defined as the minimum value of d(u) + d(v) + d(w) − |N (u) ∩ N (v) ∩ N (w)| for a set {u, v, w} of three pairwise independent vertices u, v, and w of G. Flandrin, Jung, and Li proved that any graph G of order n is Hamitonian provided G is 2-connected and σ 3 (G) ≥ n, and traceable provided σ 3 (G) ≥ n − 1. Unfortunately, we exhibit examples showing that having σ 3 (G) ≥ n − 2 is not a guarantee for G to be AP. However, we prove that G is AP provided G is 2-connected, σ 3 (G) ≥ n−2, and G has a perfect matching or quasi-perfect matching.

“…, then G is AP. Later, he [28] further showed that if G is a connected graph on n vertices with independence number at most n 2 and such that the degree sum of any pair of nonadjacent vertices is at least n − 3, then G is AP or is isomorphic to one of two exceptional graphs. Horňák, Marczyk, Schiermeyer and Woźniak [18] showed that if for a connected graph G of order n, the degree sum of any pair of nonadjacent vertices is at least n − 5, then G is AP.…”

Section: Introductionmentioning

confidence: 99%

Arbitrarily partitionable \{2K₂, C₄\}-free graphs

Liu¹,

Wu²,

Meng³

2020

A graph G = (V, E) of order n is said to be arbitrarily partitionable if for each sequence λ = (λ 1 , λ 2 ,. .. , λ p) of positive integers with λ 1 +• • •+λ p = n, there exists a partition (V 1 , V 2 ,. .. , V p) of the vertex set V such that V i induces a connected subgraph of order λ i in G for each i ∈ {1, 2,. .. , p}. In this paper, we show that a threshold graph is arbitrarily partitionable if and only if it admits a perfect matching or a near perfect matching. We also give a necessary and sufficient condition for a {2K 2 , C 4 }-free graph being arbitrarily partitionable, as an extension for a result of Broersma, Kratsch and Woeginger [Fully decomposable split graphs, European J. Combin. 34 (2013) 567-575] on split graphs.