D-degenerate Graphs Research Articles

On finding short reconfiguration sequences between independent sets

Token Sliding Optimization asks whether there exists a sequence of at most ℓ steps that transforms independent set S into T, where at each step a token slides to an unoccupied neighboring vertex (while maintaining independence). In Token Jumping Optimization, we are instead allowed to jump from a vertex to any unoccupied vertex. Both problems are known to be FPT when parameterized by ℓ on nowhere dense graphs. In this work, we show that both problems are FPT for parameter k+ℓ+d on d-degenerate graphs as well as for parameter |M|+ℓ+Δ on graphs having a modulator M to maximum degree Δ. We complement these results by showing that for parameter ℓ both problems become hard already on 2-degenerate graphs. Finally, we show that using such families one can obtain a unified algorithm for the standard Token Jumping problem parameterized by k on degenerate and nowhere dense graphs.

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and {b}-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where d≥2, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable.

The Space Complexity of Sum Labelling

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n-vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in mathscr {O}(log n) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n-vertex, m-edge, d-degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as mathscr {O}(n^2d). This enables us to store the graph using mathscr {O}(mlog n) bits of memory. For sparse graphs (graphs with mathscr {O}(n) edges), this matches the trivial lower bound of Omega (nlog n). As planar graphs and forests have constant degeneracy, our result implies an upper bound of mathscr {O}(n^2) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was mathscr {O}(4^n), due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.

Domination and convexity problems in the target set selection model

In the target set selection model (TSS), introduced by Chen in 2009, it is given a graph G with a threshold function τ:V(G)→N upper-bounded by the vertex degree. For a given model, a set S0⊆V(G) is a target set if V(G) can be partitioned into non-empty subsets S0,S1,…,St such that, for i∈{1,…,t}, Si contains exactly every vertex v outside S0∪⋯∪Si−1 having at least τ(v) neighbors in S0∪⋯∪Si−1. We say that t is the activation time tτ(S0) of the target set S0. The problem TSS-Size of finding a minimum target set has been widely studied in the literature. In 2013, Cicalese et al. investigated the problem of finding a target set with activation time 1, which we call here TSS-Domination. In 2021, Keiler et al. investigated the problem TSS-Time of finding a target set with maximum activation time. In this paper, we introduce the problem of finding a maximum proper subset of vertices which is not a target set, which we call TSS-Max-Convex. We prove that, even in split graphs and bipartite graphs, TSS-max-convex is Poly-APX-Complete, TSS-Domination is Log-APX-Complete and their parameterized versions are W[1]-hard and W[2]-hard, respectively, when parameterized by the size of the solution and the maximum threshold τ∗=maxv∈V(G)τ(v). Moreover, we prove that both problems are FPT in bounded local-treewidth graphs when parameterized by the size of the solution and the maximum threshold τ∗. Finally, we prove that TSS-Domination is FPT in d-degenerate graphs when parameterized by the size of the solution and the degeneracy d, extending the main result of Alon and Gutner in 2009.

Parameterized complexity of perfectly matched sets

For an undirected graph G, a pair of vertex disjoint subsets (A,B) is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (=|B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2O(k)⋅nO(1), and ii) Kb,b-free graphs. We obtain a linear kernel for planar graphs and kO(d)-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

On Subgraph Complementation to H-free Graphs

For a class $$\mathcal {G}$$ of graphs, the problem Subgraph Complement to $$\mathcal {G}$$ asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in $$\mathcal {G}$$ . We investigate the complexity of the problem when $$\mathcal {G}$$ is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results: Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time $$2^{o(\mid V(G)\mid )}$$ ), assuming the Exponential Time Hypothesis. We show that the complexity results on a graph class $$\mathcal {G}$$ is also true for the class $$\overline{\mathcal {G}}$$ of the complement graphs of $$\mathcal {G}$$ . Therefore, each of the above results mentioned for the H-free class of graphs is also valid for the $$\overline{H}$$ -free class of graphs. It is noteworthy that our results generalize two main results, namely, Subgraph Complement to triangle-free graphs and Subgraph Complement to d-degenerate graphs, and resolves one open question due to Fomin et al. (Algorithmica, 2020).

A polynomial version of Cereceda's conjecture

Let k and d be positive integers such that k≥d+2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper colouring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length.The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d+2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n2). So far, there is no proof of a polynomial upper bound on the diameter, even for d=2.In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(nd+1) for k≥d+2. Moreover, we prove that if k≥32(d+1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k≥2d+1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

K-Efficient domination: Algorithmic perspective

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

Fast algorithm of equitably partitioning degenerate graphs into graphs with lower degeneracy

A q-degenerate k-partition of a graph G is a collection (V1,V2,…,Vk) of k pairwise disjoint subsets of V(G) such that V(G)=⋃i=1kVi and each Vi induces a q-degenerate subgraph. Such a partition is called equitable if ||Vi|−|Vj||≤1 for every 1≤i<j≤k. Equitable partition of graphs can model the problem of partitioning a large network into smaller sub-modules based on some cardinal principles, and has many other applications in network science and information science. In this work, we establish theoretical and algorithmic results on equitably partitioning degenerate graphs into graphs with lower degeneracy. Specifically, we show that every n-vertex d-degenerate graph with maximum degree at most n/β has a q-degenerate k-partition (V1,V2,…,Vk) for every k≥αd so that each Vi has size at most ⌈n/k⌉ whenever q,α, and β satisfy a well-defined inequality, and such a partition can be computed in cubic time.

Recoloring graphs of treewidth 2

Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any (d+2)-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for d=1. Bousquet and Perarnau proved that a linear transformation exists for between (2d+2)-colorings. It is open to determine if this bound can be reduced.In this paper, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. More formally, we prove that there always exists a linear transformation between any pair of 5-colorings. This result is tight since there exist graphs of treewidth 2 and two 4-colorings such that a shortest transformation between them is quadratic.

Equitable Vertex Arboricity Conjecture Holds for Graphs with Low Degeneracy

The equitable tree-coloring can formulate a structure decomposition problem on the communication network with some security considerations. Namely, an equitable tree-k-coloring of a graph is a vertex coloring using k distinct colors such that every color class induces a forest and the sizes of any two color classes differ by at most one. In this paper, we show some theoretical results on the equitable tree-coloring of graphs by proving that every d-degenerate graph with maximum degree at most Δ is equitably tree-k-colorable for every integer k ≥ (Δ + 1)/2 provided that Δ ≥ 9.818d, confirming the equitable vertex arboricity conjecture for graphs with low degeneracy.

Approximation in (Poly-) Logarithmic Space

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for $$d\text {-}\textsc {Hitting Set}{}$$ that runs in time $$n^{{{\,\mathrm{O}\,}}{(d^2 + (d / \epsilon ))}}$$ , uses $${{\,\mathrm{O}\,}}{((d^2 + (d / \epsilon ))\log {n})}$$ bits of space, and achieves an approximation ratio of $${{\,\mathrm{O}\,}}{((d / \epsilon ) n^{\epsilon })}$$ for any positive $$\epsilon \le 1$$ and any $$d \in {\mathbb {N}}$$ . In particular, this yields a factor- $${{\,\mathrm{O}\,}}{(\log {n})}$$ approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space (for constant d). As a corollary, we obtain similar bounds for $$\textsc {Vertex Cover}{}$$ and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for $$\textsc {Vertex Cover}{}$$ on graphs with maximum degree $$\varDelta$$ , and an algorithm for computing maximal independent sets, both of which run in time $$n^{{{\,\mathrm{O}\,}}{(\varDelta )}}$$ and use $${{\,\mathrm{O}\,}}{(\varDelta \log {n})}$$ bits of space. For the more general $$d\text {-}\textsc {Hitting Set}{}$$ problem, we devise a factor-d approximation algorithm which runs in time $$n^{{{\,\mathrm{O}\,}}{(d{\delta }^2)}}$$ and uses $${{\,\mathrm{O}\,}}{(d {\delta }^2 \log {n})}$$ bits of space on set families where each element appears in at most $$\delta$$ sets. For $$\textsc {Independent Set}{}$$ restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses $${{\,\mathrm{O}\,}}{(\log {n})}$$ bits of space. We also devise a factor- $${{\,\mathrm{O}\,}}{(d^2)}$$ approximation algorithm for $$\textsc {Dominating Set}{}$$ on d-degenerate graphs which runs in time $$n^{{{\,\mathrm{O}\,}}{(\log {n})}}$$ and uses $${{\,\mathrm{O}\,}}{(\log ^2{n})}$$ bits of space. For d-regular graphs, we show how a known randomized factor- $${{\,\mathrm{O}\,}}{(\log {d})}$$ approximation algorithm can be derandomized to run in time $$n^{{{\,\mathrm{O}\,}}{(1)}}$$ and use $${{\,\mathrm{O}\,}}{(\log n)}$$ bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.

Conflict Free Version of Covering Problems on Graphs: Classical and Parameterized

Let π be a family of graphs. In the classical π-Vertex Deletion problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G − S is in π. In this paper, we introduce the conflict free version of this classical problem, namely Conflict Free π-Vertex Deletion (CF-π-VD), and study this problem from the viewpoint of classical and parameterized complexity. In the CF-π-VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set $S\subseteq V(G)$ , of size at most k, such that G − S is in π and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free versions of several classical problems. Our first result shows that if π is characterized by a finite family of forbidden induced subgraphs then CF-π-VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free versions of several well studied problems. Next, we show that if π is characterized by a “well-behaved” infinite family of forbidden induced subgraphs, then CF-π-VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF-π-VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free version of several well-known problems such as Feedback Vertex Set, Odd Cycle Transversal, Chordal Vertex Deletion and Interval Vertex Deletion are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.

On the parameterized complexity of [1,j]-domination problems

For a graph G, a set D⊆V(G) is called a [1,j]-dominating set if every vertex in V(G)∖D has at least one and at most j neighbors in D. A set D⊆V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study the aforementioned problems on various graph classes from the perspective of parameterized complexity and prove the following results:•[1,j]-Dominating Set parameterized by solution size is W[1]-hard on d-degenerate graphs for d=j+1.•[1,j]-Dominating Set parameterized by solution size is FPT on nowhere dense graphs.•The known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH).•Assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.•Finally, we study another variant of Dominating Set, called Restrained Dominating Set, that asks if there is a dominating set D of G of size at most k such that no vertex outside of D has all of its neighbors in D. We prove this variant is W[1]-hard even on 3-degenerate graphs.

Mim-width III. Graph powers and generalized distance domination problems

We generalize the family of (σ,ρ) problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as Distance-rDominating Set and Distance-rIndependent Set. We show that these distance problems are in XP parameterized by the structural parameter mim-width, and hence polynomial-time solvable on graph classes where mim-width is bounded and quickly computable, such as k-trapezoid graphs, Dilworth k-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, k-polygon graphs, circular arc graphs, complements of d-degenerate graphs, and H-graphs if given an H-representation. We obtain these results by showing that taking any power of a graph never increases its mim-width by more than a factor of two. To supplement these findings, we show that many classes of (σ,ρ) problems are W[1]-hard parameterized by mim-width + solution size.We show that powers of graphs of tree-width w−1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1.

Packing degenerate graphs

Given D and γ>0, whenever c>0 is sufficiently small and n sufficiently large, if G is a family of D-degenerate graphs of individual orders at most n, maximum degrees at most cnlog⁡n, and total number of edges at most (1−γ)(n2), then G packs into the complete graph Kn. Our proof proceeds by analysing a natural random greedy packing algorithm.

On approximate preprocessing for domination and hitting subgraphs with connected deletion sets

In this paper, we study the ConnectedH-hitting Set and Dominating Set problems from the perspective of approximate kernelization, a framework recently introduced by Lokshtanov et al. [STOC 2017]. For the ConnectedH-hitting set problem, we obtain an α-approximate kernel for every α>1 and complement it with a lower bound for the natural weighted version. We then perform a refined analysis of the tradeoff between the approximation factor and kernel size for the Dominating Set problem on d-degenerate graphs, and provide an interpolation of approximate kernels between the known 3d-approximate kernel of constant size and 1-approximate kernel of size kO(d2).

Induced and weak induced arboricities

We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G.For a class F of graphs and a graph parameter p, let p(F)=sup{p(G)∣G∈F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F)≠∞ if and only if χ(F∇12)≠∞, where F∇12 is the class of 12-shallow minors of graphs from F andχ is the chromatic number.As a main contribution of this paper, we provide bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8≤ia(F)≤10.In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.

Local antimagic orientations of [formula omitted]-degenerate graphs

A k-antimagic labeling of a digraph D with n vertices and m arcs is an injection from the set of arcs of D to the integers {1,…,m+k} such that all n vertex-sums are pairwise distinct, where the vertex-sum of a vertex v is the sum of labels of all arcs entering v minus the sum of labels of all arcs leaving v. An orientation D of a graph G is called k-antimagic if D has a k-antimagic labeling. Hefetz et al. (2010) conjectured that every connected graph admits an antimagic orientation, where “antimagic” is short for “0-antimagic”. In this paper, we consider local k-antimagic orientations of graphs. An orientation D of a graph G is called local k-antimagic if there is an injective edge labeling from E(G) to {1,…,|E(G)|+k} such that any two adjacent vertices of D have different vertex-sums. We prove that every d-degenerate graph admits a local (d+2)-antimagic orientation.

FPT algorithms for domination in sparse graphs and beyond

We study the parameterized complexity of domination problems on sparse graphs and beyond. The nowhere dense classes of graphs have been proposed as the main model for sparseness that can be utilized algorithmically. The class of d-degenerate graphs is not nowhere dense, yet domination remains fixed-parameter tractable. Both nowhere dense classes of graphs and d-degenerate graph classes are biclique-free classes, meaning there is an integer t such that no graph in the class contains Kt,t as a subgraph. In this paper we show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs. Our algorithms are simple and rely on results from extremal graph theory that bound the number of edges in a t-biclique free graph. In particular, the problems k-Dominating Set, Connectedk-Dominating Set, Independentk-Dominating Set and Minimum Weightk-Dominating Set are shown to be FPT, when parameterized by t+k, on graphs not containing Kt,t as a subgraph. With the exception of Connectedk-Dominating Set all described algorithms are linear in the size of the input graph.