Planar Graphs Of Maximum Degree Research Articles

Feedback vertex set reconfiguration in planar graphs

We study the complexity of deciding whether for two given feedback vertex sets of a graph there is a step-by-step transformation between them, such that for each feedback vertex set in the transformation, the next one is obtained by exchanging a single vertex. We give a classification of the complexity of this question for planar graphs in terms of the maximum degree. We show that for planar graphs of maximum degree at most three the problem is tractable because there always exists a transformation, while it is PSPACE-complete when the maximum degree is at most four. The positive side of the classification extends to K3,3-minor-free graphs of maximum degree three. We then consider the Matroid Parity problem, which generalizes feedback vertex sets in graphs of maximum degree three as well as matchings and spanning trees in general graphs. Generalizing known results for the latter two we show that there always exists a transformation between any two non-maximum independent parity sets.

Partitioning edges of a planar graph into linear forests and a matching

AbstractWe show that the edges of any planar graph of maximum degree at most 9 can be partitioned into four linear forests and a matching. Combined with known results, this implies that the edges of any planar graph of odd maximum degree can be partitioned into linear forests and one matching. This strengthens well‐known results stating that graphs in this class have chromatic index  and linear arboricity at most .

TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the \textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be $\mathtt{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, as a first step toward answering this question, for $k \geq 4$, we present a polynomial-time algorithm that solves \textsc{$k$-PVCR} under $\mathsf{TS}$ for caterpillars (i.e., trees formed by attaching leaves to a path).

Improved square coloring of planar graphs

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that 32Δ+1 colors are sufficient to square color every planar graph of maximum degree Δ. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that 2Δ+7 colors are sufficient, which improves the best known bounds when 6⩽Δ⩽31.

Sliding column model for t-unit bar visibility representations of graphs

A bar visibility representation of a graph [Formula: see text] is an assignment of the vertices of [Formula: see text] to distinct horizontal line segments in the plane so that two vertices are adjacent in [Formula: see text] if and only if there is an uninterrupted vertical channel of positive width that joins the bars corresponding to those vertices. A bar visibility representation is called a unit bar visibility representation if every bar has the same length. If each vertex is assigned to [Formula: see text] distinct bars of the same length in a unit bar visibility representation then the representation is called [Formula: see text]-unit bar visibility representation. In this paper, we introduce a “sliding column model” for [Formula: see text]-unit bar visibility representation and show that every graph of maximum degree [Formula: see text] has a [Formula: see text]-unit bar visibility representation for [Formula: see text]. We also show that a planar graph of maximum degree 3 having [Formula: see text] vertices and [Formula: see text] edges has a 2-unit bar visibility representation on [Formula: see text] columns and a 3-connected cubic graph of [Formula: see text] vertices admits a 2-unit bar visibility representation on [Formula: see text] columns.

Dual Circumference and Collinear Sets

We show that, if an n-vertex triangulation G of maximum degree \(\Delta \) has a dual that contains a cycle of length \(\ell \), then G has a non-crossing straight-line drawing in which some set, called a collinear set, of \(\Omega (\ell /\Delta ^4)\) vertices lie on a line. Using the current lower bounds on the length of longest cycles in cubic 3-connected graphs, this implies that every n-vertex planar graph of maximum degree \(\Delta \) has a collinear set of size \(\Omega (n^{0.8}/\Delta ^4)\).

Subcubic planar graphs of girth 7 are class I

We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974.

Non-monotone target sets for threshold values restricted to $0$, $1$, and the vertex degree

We consider a non-monotone activation process $(X_t)_{t\in\{ 0,1,2,\ldots\}}$ on a graph $G$, where $X_0\subseteq V(G)$, $X_t=\{ u\in V(G):|N_G(u)\cap X_{t-1}|\geq \tau(u)\}$ for every positive integer $t$, and $\tau:V(G)\to \mathbb{Z}$ is a threshold function. The set $X_0$ is a so-called non-monotone target set for $(G,\tau)$ if there is some $t_0$ such that $X_t=V(G)$ for every $t\geq t_0$. Ben-Zwi, Hermelin, Lokshtanov, and Newman [Discrete Optimization 8 (2011) 87-96] asked whether a target set of minimum order can be determined efficiently if $G$ is a tree. We answer their question in the affirmative for threshold functions $\tau$ satisfying $\tau(u)\in \{ 0,1,d_G(u)\}$ for every vertex~$u$. For such restricted threshold functions, we give a characterization of target sets that allows to show that the minimum target set problem remains NP-hard for planar graphs of maximum degree $3$ but is efficiently solvable for graphs of bounded treewidth.

Algorithmic aspects of broadcast independence

An independent broadcast on a connected graph G is a function f:V(G)→N0 such that, for every vertex x of G, the value f(x) is at most the eccentricity of x in G, and f(x)>0 implies that f(y)=0 for every vertex y of G within distance at most f(x) from x. The broadcast independence number αb(G) of G is the largest weight ∑x∈V(G)f(x) of an independent broadcast f on G.We describe an efficient algorithm that determines the broadcast independence number of a given tree. Furthermore, we show NP-hardness of the broadcast independence number for planar graphs of maximum degree four, and hardness of approximation for general graphs. Our results solve problems posed by Dunbar (2006), Hedetniemi (2006), and Ahmane et al. (2018).

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable

In this paper, by applying the discharging method, we show that if G is a planar graph with a maximum degree of Δ = 6 that does not contain any adjacent 8-cycles, then G is of class 1.

On the computational complexity of the bipartizing matching problem

Given a graph $$G=(V,E)$$ , an edge bipartization set of G is a subset $$E'\subseteq E(G)$$ such that $$G'=(V,E{\setminus } E')$$ is bipartite. An edge bipartization set that is also a matching of G is called a bipartizing matching of G. Let $${\mathscr {BM}}$$ be the family of all graphs admitting a bipartizing matching. Although every graph has an edge bipartization set, the problem of recognizing graphs G having bipartizing matchings ( $$G\in \mathscr {BM}$$ ) is challenging. A (k, d)-coloring of a graph G is a k-coloring of V(G) such that each vertex of G has at most d neighbors with the same color as itself. Clearly a (k, 0)-coloring is a proper vertex k-coloring of G and, for any $$d>0$$ , the k-coloring is non-proper, also known as defective. A graph belongs to $$\mathscr {BM}$$ if and only if it admits a (2, 1)-coloring. Lovász (1966) proved that for any integer $$k>0$$ , any graph of maximum degree $$\varDelta $$ admits a $$\left( k,\lfloor \varDelta /k \rfloor \right) $$ -coloring. In this paper, we show that it is NP-complete to determine whether a 3-colorable planar graph of maximum degree 4 belongs to $$\mathscr {BM}$$ , i.e., (2, 1)-colorable. Besides, we show that it is NP-complete to determine whether planar graphs of maximum degree 6 or 8 admit a (2, 2) or (2, 3)-coloring, respectively. Due to Lovász (1966), our results are tight in the sense that on graphs with maximum degree three, five, and seven, there always exists a (2, 1), (2, 2), and (2, 3)-coloring, respectively. Finally, we present polynomial-time algorithms for particular graph classes, besides some remarks on the parameterized complexity of the problem of recognizing graphs in $${\mathscr {BM}}$$ .

On the star decomposition of a graph: Hardness results and approximation for the max–min optimization problem

We study the problem of decomposing a graph into stars so that the minimum size star in the decomposition is as large as possible. Problems of graph decomposition have been actively investigated since the 70’s. The question we consider here also combines the structure of a facility location problem (choosing the centres of the stars) with a max–min fairness optimization criterion that has recently received attention in resource allocation problems, e.g., the Santa Claus problem.We focus on computational and algorithmic questions: We show that the problem is hard even in the case of planar graphs of maximum degree not larger than four, and already for decompositions into stars of size at least three. We are able to tightly characterize the boundaries between efficiently solvable instances and hard ones: we show that relaxing any of the conditions in our hardness result (minimum size of the stars or degree of the input graph) makes the problem polynomially solvable.Our complexity result implies also the APX hardness of the problem ruling out any approximation guarantee better than 2/3. We complement this inapproximability result with a 1/2-approximation algorithm. Finally, we give a polynomial time algorithm for trees. A nice property of our algorithms is that they can all be implemented to run in time linear in the size of the input graph.

Approximability of the independent feedback vertex set problem for bipartite graphs

Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed ε>0, unless P=NP, there exists no polynomial-time n1−ε-approximation algorithm even for bipartite planar graphs. We then give an α(Δ−1)/2-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in t(α,G)+O(Δn) time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in t(α,G) time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

The complexity of total k-domatic partition and total R-domination on graphs with weak elimination orderings

In this paper, we propose two linear-time algorithms. One is for computing a weak elimination ordering of a bipartite distance-hereditary graph, and the other one is an alternative algorithm to solve the total R-domination problem for any chordal bipartite graph with a weak elimination ordering. Our two linear-time algorithms lead to a unified approach to several variations of total domination problems for bipartite distance-hereditary graphs. We also show that tthe total 3-domatic partition problem is NP-complete for planar graphs of maximum degree 9 and planar bipartite graphs of maximum degree 12, and show that the 4-domatic partition problem for planar graphs of maximum degree d is polynomial-time reducible to the total 4-domatic partition problem for planar graphs of maximum degree d + 1.

Diameter of colorings under Kempe changes

Given a k-coloring of a graph G, a Kempe-change for two colors a and b produces another k-coloring of G, as follows: first choose a connected component in the subgraph of G induced by the two color classes of a and b, and then swap the colors a and b in the component. Two k-colorings are called Kempe-equivalent if one can be transformed into the other by a sequence of Kempe-changes. We consider two problems, defined as follows: First, given two k-colorings of a graph G, Kempe Reachability asks whether they are Kempe-equivalent; and second, given a graph G and a positive integer k, Kempe Connectivity asks whether any two k-colorings of G are Kempe-equivalent. We analyze the complexity of these problems from the viewpoint of graph classes. We prove that Kempe Reachability is PSPACE-complete for any fixed k≥3, and that it remains PSPACE-complete even when restricted to three colors and planar graphs of maximum degree six. Furthermore, we show that both problems admit polynomial-time algorithms on chordal graphs, bipartite graphs, and cographs. For each of these graph classes, we give a non-trivial upper bound on the number of Kempe-changes needed in order to certify that two k-colorings are Kempe-equivalent.

A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

Embedding planar 5-graphs in three pages

A book embedding of a graph G is an embedding of its vertices along the spine of a book, and an embedding of its edges to the pages so that no two edges on the same page cross. A planar graph of maximum degree k is called a planark-graph. Bekos et al. described an O(n2) time algorithm for computing two-page book embeddings for planar 4-graphs. In this paper, we embed planar 5-graphs into books of three pages by an O(n2) time algorithm.

List-edge-colouring planar graphs with precoloured edges

Let G be a simple planar graph of maximum degree Δ, let t be a positive integer, and let L be an edge list assignment on G with L(e)≥Δ+t for all e∈E(G). We prove that if H is a subgraph of G that has been L-edge-coloured, then the edge-precolouring can be extended to an L-edge-colouring of G, provided that H has maximum degree d≤t and either d≤t−4 or Δ is large enough (Δ≥16+d suffices). If d>t, there are examples for any choice of Δ where the extension is impossible.

A characterization for the neighbor-distinguishing total chromatic number of planar graphs with [formula omitted]

The neighbor-distinguishing total chromatic number χa′′(G) of a graph G is the smallest integer k such that G can be totally colored using k colors with a condition that any two adjacent vertices have different sets of colors. In this paper, we give a sufficient and necessary condition for a planar graph G with maximum degree 13 to have χa′′(G)=14 or χa′′(G)=15. Precisely, we show that if G is a planar graph of maximum degree 13, then 14≤χa′′(G)≤15; and χa′′(G)=15 if and only if G contains two adjacent 13-vertices.