Star Forest Research Articles

Facial unique-maximum colorings of plane graphs with restriction on big vertices

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and Göring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova et al. (2018). We conclude the paper by proposing some problems.

Decomposing Graphs into a Spanning Tree, an Even Graph, and a Star Forest

We prove that every connected graph can be edge-decomposed into a spanning tree, an even graph, and a star forest.

Graph 2-rankings

A 2-ranking of a graph G is an ordered partition of the vertices of G into independent sets $$V_1, \ldots , V_t$$ such that for $$i<j$$ , the subgraph of G induced by $$V_i \cup V_j$$ is a star forest in which each vertex in $$V_i$$ has degree at most 1. A 2-ranking is intermediate in strength between a star coloring and a distance-2 coloring. The 2-ranking number ofG, denoted $$\chi _{2}(G)$$ , is the minimum number of parts needed for a 2-ranking. For the d-dimensional cube $$Q_d$$ , we prove that $$\chi _{2}(Q_d) = d+1$$ . As a corollary, we improve the upper bound on the star chromatic number of products of cycles when each cycle has length divisible by 4. Let $$\chi _{2}'(G)=\chi _{2}(L(G))$$ , where L(G) is the line graph of G; equivalently, $$\chi _{2}'(G)$$ is the minimum t such that there is an ordered partition of E(G) into t matchings $$M_1, \ldots , M_t$$ such that for each j, the matching $$M_j$$ is induced in the subgraph of G with edge set $$M_1 \cup \cdots \cup M_j$$ . We show that $$\chi _{2}'(K_{m,n})=nH_m$$ when m! divides n, where $$K_{m,n}$$ is the complete bipartite graph with parts of sizes m and n, and $$H_m$$ is the harmonic sum $$1 + \cdots + \frac{1}{m}$$ . We also prove that $$\chi _{2}(G) \le 7$$ when G is subcubic and show the existence of a graph G with maximum degree k and $$\chi _{2}(G) \ge \varOmega (k^2/\log (k))$$ .

The Turán number of star forests

The Turán number of a graph H, denoted by ex(n, H), is the maximum number of edges in any graph on n vertices containing no H as a subgraph. Let Sℓ denote the star on ℓ+1 vertices and let k · Sℓ denote disjoint union of k copies of Sℓ. In this paper, for appropriately large n, we determine the Turán numbers for k · Sℓ and F, where F is a forest with components each of order 4, which improve the results of Lidický et al. (2013).

Degree powers in graphs with a forbidden forest

Given a positive integer p and a graph G with degree sequence d1,…,dn, we define ep(G)=∑i=1ndip. Caro and Yuster introduced a Turán-type problem for ep(G): Given a positive integer p and a graph H, determine the function exp(n,H), which is the maximum value of ep(G) taken over all graphs G on n vertices that do not contain H as a subgraph. Clearly, ex1(n,H)=2ex(n,H), where ex(n,H) denotes the classical Turán number. Caro and Yuster determined the function exp(n,Pℓ) for sufficiently large n, where p≥2 and Pℓ denotes the path on ℓ vertices. In this paper, we generalise this result and determine exp(n,F) for sufficiently large n, where p≥2 and F is a linear forest. We also determine exp(n,S), where S is a star forest; and exp(n,B), where B is a broom graph with diameter at most six.

Complexity and approximability of extended Spanning Star Forest problems in general and complete graphs

A solution extension problem consists in an instance and a partial feasible solution which is given in advance and the goal is to extend this partial solution to a feasible one. Many well-known problems like Coloring Extension Problems, Traveling Salesman Problem and Routing problems, have been studied in this framework. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP-hardness, parameterized complexity, positive and negative approximation results.

Thin trees in 8-edge-connected planar graphs

Merker and Postle showed that any graph that can be decomposed into a forest and a star forest has a decomposition into two forests with diameter at most 18. Using this result, they proved any planar graph of girth at least 6 has two edge-disjoint 1819-thin spanning trees. By using a simpler version of their techniques, given a graph that can be decomposed into a forest and a matching, in polynomial time, we decompose the graph into two forests with diameter at most 12. Using this result, we are able to show that any 8-edge-connected planar graph has two edge-disjoint 1213-thin spanning trees.

Bounded diameter arboricity

AbstractWe introduce the notion of bounded diameter arboricity. Specifically, the diameter‐ arboricity of a graph is the minimum number such that the edges of the graph can be partitioned into forests each of whose components has diameter at most . A class of graphs has bounded diameter arboricity if there exists a natural number such that every graph in the class has diameter‐ arboricity at most . We conjecture that the class of graphs with arboricity at most has bounded diameter arboricity at most . We prove this conjecture for by proving the stronger assertion that the union of a forest and a star forest can be partitioned into two forests of diameter at most 18. We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least for all . As an application we show that every 6‐edge‐connected planar (multi)graph contains two edge‐disjoint ‐thin spanning trees. Moreover, we answer a question by Mütze and Peter, leading to an improved lower bound on the density of globally sparse Ramsey graphs.

On Star Forest Ascending Subgraph Decomposition

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.

Stability in the Erdős–Gallai Theorems on cycles and paths

The Erdős–Gallai Theorem states that for k≥2, every graph of average degree more than k−2 contains a k-vertex path. This result is a consequence of a stronger result of Kopylov: if k is odd, k=2t+1≥5, n≥(5t−3)/2, and G is an n-vertex 2-connected graph with at least h(n,k,t):=(k−t2)+t(n−k+t) edges, then G contains a cycle of length at least k unless G=Hn,k,t:=Kn−E(Kn−t).In this paper we prove a stability version of the Erdős–Gallai Theorem: we show that for all n≥3t>3, and k∈{2t+1,2t+2}, every n-vertex 2-connected graph G with e(G)>h(n,k,t−1) either contains a cycle of length at least k or contains a set of t vertices whose removal gives a star forest. In particular, if k=2t+1≠7, we show G⊆Hn,k,t. The lower bound e(G)>h(n,k,t−1) in these results is tight and is smaller than Kopylov's bound h(n,k,t) by a term of n−t−O(1).

I,F-partitions of sparse graphs

A stark-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than 52 has an I,F-partition, which is sharp and answers a question of Cranston and West (0000). This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu et al. (2009).

Three ways to cover a graph

We consider the problem of covering an input graphH with graphs from a fixed covering classG. The classical covering number of H with respect to G is the minimum number of graphs from G needed to cover the edges of H without covering non-edges of H. We introduce a unifying notion of three covering parameters with respect to G, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures “how far” H is from G in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention.We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class.

The maximum weight spanning star forest problem on cactus graphs

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946–962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus.

On the ascending subgraph decomposition problem for bipartite graphs

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1⊕⋯⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i=1,…,n−1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies di≥n−i+2, 1≤i<k, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.

Partitioning the arcs of a digraph into a star forest of the underlying graph with prescribed orientation properties

A star in an undirected graph is a tree in which at most one vertex has degree larger than one. A star forest is a collection of vertex disjoint stars. An out-star (in-star) in a digraph D is a star in the underlying undirected graph of D such that all edges are directed out of (into) the center. The problem of partitioning the edges of the underlying graph of a digraph D into two star forests F0 and F1 is known to be NP-complete. On the other hand, with the additional requirement for F0 and F1 to be forests of out-stars the problem becomes polynomial (via an easy reduction to 2-SAT). In this article we settle the complexity of problems lying in between these two problems. Namely, we study the complexity of the related problems where we require each Fi to be a forest of stars in the underlying sense and require (in different problems) that in D, Fi is either a forest of out-stars, in-stars, out- or in-stars or just stars in the underlying sense.

APPROXIMATING THE SPANNING k-TREE FOREST PROBLEM

The spanning star forest problem is an interesting algorithmic problem in combinatorial optimization and finds different applications. We generalize it into the spanning k-tree forest problem, which is to find a maximum spanning forest in which each tree component has a central vertex and other vertices in the component have distance at most k away from the central vertex. We show that this new problem can be approximated with ratio [Formula: see text] in polynomial time for both undirected and directed graphs. In the weighted distance model, a ½-approximation algorithm is presented for it.

Star list chromatic number of planar subcubic graphs

A proper coloring of the vertices of a graph G is called a star-coloring if the union of every two color classes induces a star forest. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring ? such that ?(v)?L(v). If G is L-star-colorable for any list assignment L with |L(v)|?k for all v?V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by $\chi_{s}^{l}(G)$ , is the smallest integer k such that G is k-star-choosable. In this paper, we prove that every planar subcubic graph is 6-star-choosable.

Edge covering pseudo-outerplanar graphs with forests

A graph is pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or two forests and a matching, or max{Δ(G),4} matchings, or max{⌈Δ(G)/2⌉,3} linear forests. These results generalize known results on outerplanar graphs and K2,3-minor-free graphs, since the class of pseudo-outerplanar graphs is larger than the class of K2,3-minor-free graphs.

Improved approximation for spanning star forest in dense graphs

A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c?(0,1), a graph of n vertices is called c-dense if it contains at least cn 2/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ?>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c?(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.

Improved Approximation Algorithms for the Spanning Star Forest Problem

A star graph is a tree of diameter at most two. A star forest is a graph that consists of node-disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set.We present a 0.71-approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946–962, 2008). We also present a 0.64-approximation algorithm for the problem on node-weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge-weighted and node-weighted) versions of the problem.Our algorithms use a non-linear rounding scheme, which might be of independent interest.