On-line arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.     

Dense Arbitrarily Vertex Decomposable Graphs

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n ₁, . . . , n _k ) of positive integers such that n ₁ + · · · + n _k = n there exists a partition (V ₁, . . . , V _k ) of the vertex set of G such that for each ${i \in \{1,\ldots,k\}}$ , V _i induces a connected subgraph of G on n _i vertices. The main result of the paper reads as follows. Suppose that G is a connected graph on n ≥ 20 vertices that admits a perfect matching or a matching omitting exactly one vertex. If the degree sum of any pair of nonadjacent vertices is at least n ? 5, then G is arbitrarily vertex decomposable. We also describe 2-connected arbitrarily vertex decomposable graphs that satisfy a similar degree sum condition.     

An Ore-type condition for arbitrarily vertex decomposable graphs

Let G be a graph of order n and r, 1≤r≤n, a fixed integer. G is said to be r-vertex decomposable if for each sequence (n₁,…,n_r) of positive integers such that n₁+?+n_r=n there exists a partition (V₁,…,V_r) of the vertex set of G such that for each i∈{1,…,r}, V_i induces a connected subgraph of G on n_i vertices. G is called arbitrarily vertex decomposable if it is r-vertex decomposable for each r∈{1,…,n}.In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n−3, then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r-vertex decomposable.     

Arbitrarily vertex decomposable suns with few rays

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,…,n_k) of positive integers with n₁+?+n_k=n, there exists a partition (V₁,…,V_k) of the vertex set of G such that V_i induces a connected subgraph of order n_i, for all i=1,…,k. A sun with r rays is a unicyclic graph obtained by adding r hanging edges to r distinct vertices of a cycle. We characterize all arbitrarily vertex decomposable suns with at most three rays. We also provide a list of all on-line arbitrarily vertex decomposable suns with any number of rays.     

Recognizing decomposable graphs

A graph is called decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v. We point out a simple and efficient algorithm for recognizing decomposable graphs with maximum degree at most 3 and prove that recognizing decomposable graphs with maximum degree 4 is an NP-complete problem.     

On arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ; from a result by Barth and Fournier it follows that Δ(T)?4. A necessary and a sufficient condition for being an arbitrarily vertex decomposable star-like tree have been exhibited. The conditions seem to be very close to each other.     

On-line secret sharing

In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified subsets have no information on the secret. In an on-line secret sharing scheme the dealer assigns shares in the order the participants show up, knowing only those qualified subsets whose all members she has seen. We often assume that the overall access structure (the set of minimal qualified subsets) is known and only the order of the participants is unknown. On-line secret sharing is a useful primitive when the set of participants grows in time, and redistributing the secret when a new participant shows up is too expensive. In this paper we start the investigation of unconditionally secure on-line secret sharing schemes. The complexity of a secret sharing scheme is the size of the largest share a single participant can receive over the size of the secret. The infimum of this amount in the on-line or off-line setting is the on-line or off-line complexity of the access structure, respectively. For paths on at most five vertices and cycles on at most six vertices the on-line and offline complexities are equal, while for other paths and cycles these values differ. We show that the gap between these values can be arbitrarily large even for graph based access structures. We present a general on-line secret sharing scheme that we call first-fit. Its complexity is the maximal degree of the access structure. We show, however, that this on-line scheme is never optimal: the on-line complexity is always strictly less than the maximal degree. On the other hand, we give examples where the first-fit scheme is almost optimal, namely, the on-line complexity can be arbitrarily close to the maximal degree. The performance ratio is the ratio of the on-line and off-line complexities of the same access structure. We show that for graphs the performance ratio is smaller than the number of vertices, and for an infinite family of graphs the performance ratio is at least constant times the square root of the number of vertices.     

On the shape of decomposable trees

A n-vertex graph is said to be decomposable if, for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. The aim of the paper is to study the homeomorphism classes of decomposable trees. More precisely, we show that homeomorphism classes containing decomposable trees with an arbitrarily large minimal distance between all pairs of distinct vertices of degree different from 2, is exactly the set of combs.     

Hamiltonian decomposition of lexicographic product

In this paper we prove the conjecture of J.-C. Bermond (Ann. Discrete Math.36 (1978), 21–28): If two graphs are decomposable into Hamiltonian cycles, then their lexicographic product is decomposable, too.     

Hamilton decompositions of some line graphs

The main result of this paper completely settles Bermond's conjecture for bipartite graphs of odd degree by proving that if G is a bipartite (2k + 1)-regular graph that is Hamilton decomposable, then the line graph, L(G), of G is also Hamilton decomposable. A similar result is obtained for 5-regular graphs, thus providing further evidence to support Bermond's conjecture. © 1995 John Wiley & Sons, Inc.     

A new view toward vertex decomposable graphs

In this paper, we bring a new view about closed neighbourhood to show the vertex decomposability of graphs. Making use of the characterization of hereditary vertex decomposable graphs, we introduce a class of vertex decomposable graphs, which include some well-known classic vertex decomposable graphs such as clique-whiskered graphs and Cameron-Walker graphs.     

Arbitrarily traceable graphs and digraphs

The work in this paper extends and generalizes earlier work by Ore on arbitrarily traceable Euler graphs, by Harary on arbitrarily traceable digraphs, by Chartrand and White on randomly n-traversable graphs, and by Chartrand and Lick on randomly Eulerian digraphs. Arbitrarily traceable graphs of mixed type are defined and characterized in terms of a class of forbidden graphs. Arbitrarily traceable digraphs of mixed type are also defined and a simply applied characterization is given for them.     

A degree bound on decomposable trees

A n-vertex graph is said to be decomposable if for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polynomial time algorithm to decide if a multipode (a tree with only one vertex of degree greater than 2) is decomposable. We also exhibit two families of decomposable trees: arbitrary large trees with one vertex of degree 4, and trees with an arbitrary number of degree-3 vertices.     

Edge-disjoint Open Trails in Complete Bipartite Multigraphs

Let ^r K _a,b be a complete bipartite multigraph. We show a necessary and sufficient condition for a multigraph ^r K _a,b to be arbitrarily decomposable into open trails. We extend the results obtained by Balister for complete graphs (Balister in Probab Comput 10:463–499, 2001). Moreover we show that a multigraph ^r K _a,b with even r and a ≥ 3 or b ≥ 3 is arbitrarily decomposable into open and closed trails.     

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition \(E_1 \cup E_2 \cup \cdots \cup E_k\) of the edge set E(G) such that each \(E_i\) induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.     

Hamiltonian decompositions of products of cycles

A graph is said to be decomposable into hamiltonian cycles if its edge set can be partitioned into hamiltonian cycles. We show that the cartesian product of any three cycles can be decomposed into three hamiltonian cycles, thus settling a conjecture by Kotzig. We also show that, more generally, the cartesian product of 2^a3^b graphs, each decomposable into m hamiltonian cycles, can be decomposed into 2^a3^bm hamiltonian cycles.     

Shedding vertices of vertex decomposable well-covered graphs

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen–Macaulay graphs.     

Combinatorial optimization with 2-joins

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset.We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.     

Partitions of graphs into cographs

Cographs form the minimal family of graphs containing K₁ that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs.     

Uniquely Hamiltonian Graphs of Minimum Degree 4

We construct an infinite family of uniquely hamiltonian graphs of minimum degree 4, maximum degree 14, and of arbitrarily high maximum degree.