Spanning trees without adjacent vertices of degree 2

Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2.     

The firefighter problem for cubic graphs

We show that the firefighter problem is NP-complete for cubic graphs. We also show that given a rooted tree of maximum degree three in which every leaf is at the same distance from the root, it is NP-complete to decide whether or not there is a strategy that protects every leaf from the fire, which starts at the root. By contrast, we describe a polynomial time algorithm to decide if it is possible to save a given subset of the vertices of a graph with maximum degree three, provided the fire breaks out at a vertex of degree at most two.     

On the elusiveness of Hamiltonian property

1. IntroductionA gash G is an ordered pair of disjoillt sets (V, E) such that E is a subset of the setof unordered pairs of V, where the sets V and E are finite. The set V is cajled the setof venices and E is called the set of edges. They are usually denoted by V(G) and E(C),respectively. An edge (x, y) is said to join the venices x and y, and is sometimes denotedby xo or ear. By our definition, a graph does not colltain any loOP, neither does it colltainmultiple edges.Other terms undef…     

A condition for assured 3-face-colorability of infinite plane graphs with a given spanning tree

Given an infinite leafless tree drawn on the plane, we ask whether or not one can add edges between the vertices of the tree obtaining a non-3-face-colorable graph. We formulate a condition conjectured to be necessary and sufficient for this to be possible. We prove that this condition is indeed necessary and sufficient for trees with maximal degree 3, and that it is sufficient for general trees. In particular, we prove that every infinite plane graph with a spanning binary tree is 3-face-colorable.     

Planar segment visibility graphs

Given a set of n disjoint line segments in the plane, the segment visibility graph is the graph whose 2n vertices correspond to the endpoints of the line segments and whose edges connect every pair of vertices whose corresponding endpoints can see each other. In this paper we characterize and provide a polynomial time recognition algorithm for planar segment visibility graphs. Actually, we characterize segment visibility graphs that do not contain the complete graph K₅ as a minor, and show that this class is the same as the class of planar segment visibility graphs. We use and prove the fact that every segment visibility graph contains K₄ as a subgraph. In fact, we prove a stronger result: every set of n line segments determines at least n−3 empty convex quadrilaterals.     

Clique trees of chordal graphs: leafage and 3-asteroidals

Chordal graphs were characterized as those graphs having a tree, called clique tree, whose vertices are the cliques of the graph and for every vertex in the graph, the set of cliques that contain it form a subtree of clique tree. In this work, we study the relationship between the clique trees of a chordal graph and its subgraphs. We will prove that clique trees can be described locally and all clique trees of a graph can be obtained from clique trees of subgraphs. In particular, we study the leafage of chordal graphs, that is the minimum number of leaves among the clique trees of the graph. It is known that interval graphs are chordal graphs without 3-asteroidals. We will prove a generalization of this result using the framework developed in the present article. We prove that in a clique tree that realizes the leafage, for every vertex of degree at least 3, and every choice of 3 branches incident to it, there is a 3−asteroidal in these branches.     

Every DFS Tree of a 3‐Connected Graph Contains a Contractible Edge

Tutte proved that every 3‐connected graph G on more than 4 vertices contains a contractible edge. We strengthen this result by showing that every depth‐first‐search tree of G contains a contractible edge. Moreover, we show that every spanning tree of G contains a contractible edge if G is 3‐regular or if G does not contain two disjoint pairs of adjacent degree‐3 vertices.     

Minimal configuration trees

A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and edges, that has nullity one and the same core as G, is called a minimal configuration. A subdivision of a graph G is obtained by inserting a vertex on every edge of G. We review various properties of minimal configurations. In particular, we show that a minimal configuration is a tree if and only if it is a subdivision of some other tree.     

Minimal configuration trees

A graph is singular of nullity η if zero is an eigenvalue of its adjacency matrix with multiplicity η. If η(G)=1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the zero-eigenvector. A connected subgraph of G with the least number of vertices and edges, that has nullity one and the same core as G, is called a minimal configuration. A subdivision of a graph G is obtained by inserting a vertex on every edge of G. We review various properties of minimal configurations. In particular, we show that a minimal configuration is a tree if and only if it is a subdivision of some other tree.     

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.     

Closest 4-leaf power is fixed-parameter tractable

The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r.     

The Erdös-Sós conjecture for spiders

A classical result on extremal graph theory is the Erdös-Gallai theorem: if a graph on n vertices has more than edges, then it contains a path of k edges. Motivated by the result, Erdös and Sós conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. In this paper, we prove that if a graph on n vertices has more than edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Wo?niak on spiders of diameter at most 4.     

List precoloring extension in planar graphs

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thomassen’s theorem can be extended by precoloring some vertices which are at a large enough distance apart in a graph. Here, among others, we answer the question in the case when the graph does not contain short cycles separating precolored vertices and when there is a “wide” Steiner tree containing all the precolored vertices.     

On the Metric Dimension of Infinite Graphs

A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we undertake the metric dimension of infinite locally finite graphs, i.e., those infinite graphs such that all its vertices have finite degree. We give some necessary conditions for an infinite graph to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some general results about the metric dimension of the Cartesian product of finite and infinite graphs, and obtain the metric dimension of the Cartesian product of several families of graphs.     

Semiregular trees with minimal Laplacian spectral radius

A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence.     

The Four-in-a-Tree Problem in Triangle-Free Graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n ⁴) whether three given vertices of a graph belong to an induced tree. Here, we study four-in- a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the “same structure”, in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.     

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.     

Acrylic improper colorings of graphs

In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class V_i satisfies some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999     

On the reconstruction of locally finite trees

We prove a theorem saying, when taken together with previous results of Bondy, Hemminger, and Thomassen, that every locally finite, infinite tree not containing a subdivision of the dyadic tree (i. e., the regular tree of degree 3) is uniquely determined, up to isomorphism, from its collection of vertex-deleted subgraphs. Furthermore, as another partial result concerning the reconstruction of locally finite trees, we show that the same is true for locally finite trees whose set of vertices of degree s is nonempty and finite (for some positive integer s).     

Special monochromatic trees in two-colored complete graphs

For a positive integer k, a set of k + 1 vertices in a graph is a k-cluster if the difference between degrees of any two of its vertices is at most k − 1. Given any tree T with at least k³ edges, we show that for each graph G of sufficiently large order, either G or its complement contains a copy of T such that some vertices in the copy form a k-cluster in G. The same conclusion holds for any tree T having a vertex of degree more than k. © 1997 John Wiley & Sons, Inc.