The Complexity of Toughness in Regular Graphs

2008

We study variants and generalizations of the problem of finding an r-regular subgraph (where r ≥ 3) in a given graph by deleting at most k vertices. have shown that the problem is fixed-parameter tractable (FPT) if parameterized by (k, r). They asked whether the problem remains fixed-parameter tractable if parameterized by k alone. We answer this question negatively: we show that if parameterized by k alone the problem is W [1]-hard and therefore very unlikely fixed-parameter tractable. We also give W [1]-hardness results for variants of the problem where the parameter is the number of vertex and edge deletions allowed, and for a new generalized form of the problem where the obtained subgraph is not necessarily regular but its vertices have certain prescribed degrees. Following this we demonstrate fixed-parameter tractability for the considered problems if the parameter includes the regularity r or an upper bound on the prescribed degrees in the generalized form of the problem. These FPT results are obtained via kernelization, so also provide a practical approach to the problems presented.

Discrete Mathematics, 1973

The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t 0 , every t 0 -tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t 0 = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of (3/2)-tough nonhamiltonian graphs.

Graphs and Combinatorics, 1999

A graph is rY k-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. with an r-regular spanning subgraph). We show that every ttough graph of order n with t 2 is 2Y k-factor-critical for every non-negative integer k minf2t À 2Y n À 3g, thus proving a conjecture as well as generalizing the main result of Liu and Yu in .

arXiv (Cornell University), 2023

The closure of a graph G is the graph G * obtained from G by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least n, where n is the number of vertices of G. The well-known Closure Lemma proved by Bondy and Chvátal states that a graph G is Hamiltonian if and only if its closure G * is. This lemma can be used to prove several classical results in Hamiltonian graph theory. We prove a version of the Closure Lemma for tough graphs. A graph G is t-tough if for any set S of vertices of G, the number of components of G − S is at most |S| t. A Hamiltonian graph must necessarily be 1-tough. Conversely, Chvátal conjectured that there exists a constant t such that every t-tough graph is Hamiltonian. The t-closure of a graph G is the graph G t * obtained from G by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least n − t. We prove that, for t ≥ 2, a 3t−1 2-tough graph G is Hamiltonian if and only if its t-closure G t * is. Hoàng conjectured the following: Let G be a graph with degree sequence d 1 ≤ d 2 ≤. .. ≤ d n ; then G is Hamiltonian if G is t-tough and, for all i < n 2 , if d i ≤ i then d n−i+t ≥ n − i. This conjecture is analogous to the well known theorem of Chvátal on Hamiltonian ideals. Hoàng proved the conjecture for t ≤ 3. Using the closure lemma for tough graphs, we prove the conjecture for t = 4.

2020

We show that there is no subexponential time algorithm for computing the exact solution of the maximum independent set problem in $d$-regular graphs, for any constant $d&gt;2$, unless ETH fails. We also discuss the extensions of our construction to other problems and other classes of graphs, including $5$-regular planar graphs.

HAL (Le Centre pour la Communication Scientifique Directe), 2015

Let G denote a multigraph with edge set E(G), let µ(G) denote the maximum edge multiplicity in G, and let P k denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if |E(G)| is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with µ(G) ≤ 2 if and only if no 3 vertices of G induce more than 4 edges and |E(G)| is divisible by 3. Oksimets (2003) proved that for all integers k ≥ 3, P4 decomposes a connected 2k-regular graph G if and only if |E(G)| is divisible by 3. We prove that for all integers k ≥ 2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k ≥ 1, every 6k-regular multigraph with µ(G) ≤ 2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to µ(G) by proving that for all integers k ≥ 3 the problem of determining if P4 decomposes a 2k-regular multigraph with µ(G) ≤ 2k 3 + 1 is NP-Complete.

Graphs and Combinatorics, 1999

A graph is rY k-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. an r-regular spanning subgraph). Let tG denote the toughness of graph G. In this paper, we show that if tG 4, then G is 3Y k-factor-critical for every non-negative integer k such that n k even, k`2tG À 2 and k n À 7.

Electronic Notes in Discrete Mathematics, 2002

We now know that not every 2-tough graph is hamiltonian. In fact for every > 0, there exists a (9/4− ) -tough nontraceable graph. We continue our quadrennial survey of results that relate the toughness of a graph to its cycle structure.

Discrete Applied Mathematics, 1998

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given graph has a stable set whose deletion results in (3) the complement of a bipartite graph, (4) a split graph or (5) a threshold graph. 0 1998 Elsevier Science B.V. All rights reserved.