Disjoint Vertex Sets Research Articles

Pure pairs. III. Sparse graphs with no polynomial‐sized anticomplete pairs

AbstractA graph is ‐free if it has no induced subgraph isomorphic to , and |G| denotes the number of vertices of . A conjecture of Conlon, Sudakov and the second author asserts that: For every graph , there exists such that in every ‐free graph with |G| there are two disjoint sets of vertices, of sizes at least and , complete or anticomplete to each other. This is equivalent to: The “sparse linear conjecture”: For every graph , there exists such that in every ‐free graph with , either some vertex has degree at least , or there are two disjoint sets of vertices, of sizes at least and , anticomplete to each other. We prove a number of partial results toward the sparse linear conjecture. In particular, we prove it holds for a large class of graphs , and we prove that something like it holds for all graphs . More exactly, say is “almost‐bipartite” if is triangle‐free and can be partitioned into a stable set and a set inducing a graph of maximum degree at most one. (This includes all graphs that arise from another graph by subdividing every edge at least once.) Our main result is: The sparse linear conjecture holds for all almost‐bipartite graphs . (It remains open when is the triangle .) There is also a stronger theorem: For every almost‐bipartite graph , there exist such that for every graph with and maximum degree less than , and for every with , either contains induced copies of , or there are two disjoint sets with and , and with at most edges between them. We also prove some variations on the sparse linear conjecture, such as: For every graph , there exists such that in every ‐free graph with vertices, either some vertex has degree at least , or there are two disjoint sets of vertices with , anticomplete to each other.

On 2-rainbow domination number of functigraph and its complement

Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp.

Computation in Causal Graphs

The goal of this paper is to introduce some of the important concepts in causal graph theory and examine them from combinatorial and computational perspectives. Of fundamental importance in applications of causal models that use graphs are dependence-separators or, simply, $d$-separators. A vertex set $Z$ is a $d$-separator for a pair of disjoint vertex sets $(X,Y)$ if $X$ and $Y$ are independent conditioned on $Z$. For the case of a single-setpair it is known that $d$-separators can be found efficiently by elegant network flow techniques. In this paper, we consider $d$-separators for a collection $\{(X_1,Y_1),(X_2,Y_2),\dots,(X_k,Y_k)\}$ of setpairs. We focus on two classes of combinatorial objects in this multiple-setpair framework: $d$-separators and $d$-super-separators. We say that $Z$ is a $d$-separator for multiple setpairs if, for each $i$, $X_i$ and $Y_i$ are independent conditioned on $Z$; we say that $Z$ is a $d$-super-separator if, for each $i$, there exists $Z_i\subseteq Z$, such that $X_i$ and $Y_i$ are independent conditioned on $Z_i$. For the latter object, we give an $O(\log^2k)$-approximation algorithm for the problem of finding a minimum cost $d$-super-separator. The focus on approximation algorithms is necessary as we show this problem is NP-complete. For the former object, we show it is hard to determine whether a $d$-separator exists, even when there are just five setpairs. This problem remains hard even if each setpair consists of singleton vertices, provided the number of setpairs is large. On the positive side, we show that if there are a fixed number of singleton setpairs then a $d$-separator for multiple setpairs can be found in polynomial time, if one exists.

The spectra of a new join of graphs

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

The upper domatic number of a graph

Let G=(V,E) be a graph. For two disjoint sets of vertices R and S, set R dominates set S if every vertex in S is adjacent to at least one vertex in R. In this paper we introduce the upper domatic numberD(G), which equals the maximum order k of a vertex partition π={V1,V2,…,Vk} such that for every i,j, 1≤i<j≤k, either Vi dominates Vj or Vj dominates Vi, or both. We study properties of the upper domatic number of a graph, determine bounds on D(G), and compare D(G) to a related parameter, the transitivity Tr(G) of G.

A linear-time algorithm for finding a one-to-many 3-disjoint path cover in the cube of a connected graph

Given two disjoint vertex sets S={s} and T={t1,t2,t3} of a connected graph, a one-to-many 3-disjoint path cover joining S and T is a vertex-disjoint path cover {P1,P2,P3} such that each path Pi joins s and ti. In this paper, we present an efficient algorithm that builds, if exists, a one-to-many 3-disjoint path cover in the cube of a connected graph G joining the two terminal sets. Interestingly enough, we show that a carefully selected spanning tree or spanning unicyclic subgraph of G is all we need to find the desired disjoint path cover in time linear to the size of G.

Hamilton Cycles, Minimum Degree, and Bipartite Holes

AbstractWe present a tight extremal threshold for the existence of Hamilton cycles in graphs with large minimum degree and without a large “bipartite hole” (two disjoint sets of vertices with no edges between them). This result extends Dirac's classical theorem, and is related to a theorem of Chvátal and Erdős. In detail, an ‐bipartite‐hole in a graph G consists of two disjoint sets of vertices S and T with and such that there are no edges between S and T; and is the maximum integer r such that G contains an ‐bipartite‐hole for every pair of nonnegative integers s and t with . Our central theorem is that a graph G with at least three vertices is Hamiltonian if its minimum degree is at least . From the proof we obtain a polynomial time algorithm that either finds a Hamilton cycle or a large bipartite hole. The theorem also yields a condition for the existence of k edge‐disjoint Hamilton cycles. We see that for dense random graphs , the probability of failing to contain many edge‐disjoint Hamilton cycles is . Finally, we discuss the complexity of calculating and approximating .

King-Serf Duo by Monochromatic Paths in $k$-Edge-Coloured Tournaments

An open conjecture of Erdős states that for every positive integer $k$ there is a (least) positive integer $f(k)$ so that whenever a tournament has its edges colored with $k$ colors, there exists a set $S$ of at most $f(k)$ vertices so that every vertex has a monochromatic path to some point in $S$. We consider a related question and show that for every (finite or infinite) cardinal $\kappa&gt;0$ there is a cardinal $ \lambda_\kappa $ such that in every $\kappa$-edge-coloured tournament there exist disjoint vertex sets $K,S$ with total size at most $ \lambda_\kappa$ so that every vertex $ v $ has a monochromatic path of length at most two from $K$ to $v$ or from $v$ to $S$.

Splice Graphs and their Vertex-Degree-Based Invariants

Let G_1 and G_2 be simple connected graphs with disjoint vertex sets V(G_1) and V(G_2), respectively. For given vertices a_1in V(G_1) and a_2in V(G_2), a splice of G_1 and G_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the union of G_1 and G_2. In this paper, we present exact formulas for computing some vertex-degree-based graph invariants of splice of graphs.

Paired many-to-many disjoint path covers in restricted hypercube-like graphs

Given two disjoint vertex-sets, S={s1,…,sk} and T={t1,…,tk} in a graph, a paired many-to-many k-disjoint path cover between S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which each path Pi runs from si to ti. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most non-bipartite hypercube-like networks found in the literature, such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we show that every m-dimensional restricted hypercube-like graph, m≥5, with at most f vertex and/or edge faults being removed has a paired many-to-many k-disjoint path cover between arbitrary disjoint sets S and T of size k each, subject to k≥2 and f+2k≤m+1. The bound m+1 on f+2k is the best possible.

Algorithms for finding disjoint path covers in unit interval graphs

A many-to-many k-disjoint path cover (k-DPC for short) of a graph G joining the pairwise disjoint vertex sets S and T, each of size k, is a collection of k vertex-disjoint paths between S and T, which altogether cover every vertex of G. This is classified as paired, if each vertex of S must be joined to a specific vertex of T, or unpaired, if there is no such constraint. In this paper, we develop a linear-time algorithm for the Unpaired DPC problem of finding an unpaired DPC joining S and T given in a unit interval graph, to which the One-to-One DPC and the One-to-Many DPC problems are reduced in linear time. Additionally, we show that the Paired k-DPC problem on a unit interval graph is polynomially solvable for any fixed k.

Unpaired many-to-many disjoint path covers in restricted hypercube-like graphs

For two disjoint vertex-sets, S={s1,…,sk} and T={t1,…,tk} of a graph, an unpaired many-to-many k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P1,…,Pk} that altogether cover every vertex of the graph, in which Pi is a path from si to some tj for 1≤i,j≤k. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most nonbipartite hypercube-like networks found in the literature, such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we show that every m-dimensional restricted hypercube-like graph, m≥5, with at most f faulty vertices and/or edges being removed has an unpaired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each subject to k≥2 and f+k≤m−1. The bound m−1 on f+k is the maximum possible.

On spectra of variants of the corona of two graphs and some new equienergetic graphs

Let G and H be two graphs. The join G ∨ H is the graph obtained by joining every vertex of G with every vertex of H. The corona G ◦ H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the i-th vertex of G to every vertex in the i-th copy of H. The neighborhood corona G⋆H is the graph obtained by taking one copy of G and |V (G)| copies of H and joining the neighbors of the i-th vertex of G to every vertex in the i-th copy of H. The edge corona G ⋄H is the graph obtained by taking one copy of G and |E(G)| copies of H and joining each terminal vertex of i-th edge of G to every vertex in the i-th copy of H. Let G1, G2, G3 and G4 be regular graphs with disjoint vertex sets. In this paper we compute the spectrum of (G1 ∨ G2) ∪ (G1 ⋆ G3), (G1 ∨ G2) ∪ (G2 ⋆ G3) ∪ (G1 ⋆ G4), (G1∨G2)∪(G1 ◦G3), (G1∨G2)∪(G2 ◦G3)∪(G1 ◦G4), (G1∨G2)∪(G1 ⋄G3), (G1 ∨ G2) ∪ (G2 ⋄ G3) ∪ (G1 ⋄ G4), (G1 ∨ G2) ∪ (G2 ◦ G3) ∪ (G1 ⋆ G3), (G1 ∨ G2) ∪ (G2 ◦ G3) ∪ (G1 ⋄ G4) and (G1 ∨ G2) ∪ (G2 ⋆ G3) ∪ (G1 ⋄ G4). As an application, we show that there exist some new pairs of equienergetic graphs on n vertices for all n ≥ 11.

The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

We derive tight bounds for the maximum number of k-faces, $$0\le k\le d-1$$0≤k≤d-1, of the Minkowski sum, $$P_1+P_2$$P1+P2, of two d-dimensional convex polytopes $$P_1$$P1 and $$P_2$$P2, as a function of the number of vertices of the polytopes. For even dimensions $$d\ge 2$$d?2, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are cyclic d-polytopes with disjoint vertex sets. For odd dimensions $$d\ge 3$$d?3, the maximum values are attained when $$P_1$$P1 and $$P_2$$P2 are $$\lfloor \frac{d}{2}\rfloor $$?d2?-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.

Highly linked tournaments

A (possibly directed) graph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi goes from xi to yi. A theorem of Bollobás and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás–Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments—there is a function f(k) such that every strongly f(k)-connected tournament is k-linked. The bound on f(k) was reduced to O(klog⁡k) by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked.

Many-to-many two-disjoint path covers in cylindrical and toroidal grids

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex sets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. The many-to-many k-disjoint path cover is classified as paired if each source in S is further required to be paired with a specific sink in T, or unpaired otherwise. In this paper, we first establish a necessary and sufficient condition for a bipartite cylindrical grid to have a paired many-to-many 2-disjoint path cover joining S and T. Based on this characterization, we then prove that, provided the set S∪T contains the equal numbers of vertices from different parts of the bipartition, the bipartite cylindrical grid always has an unpaired many-to-many 2-disjoint path cover. Additionally, we show that such balanced vertex sets also guarantee the existence of a paired many-to-many 2-disjoint path cover for any bipartite toroidal grid even if an arbitrary edge is removed.

On the complexity of the black-and-white coloring problem on some classes of perfect graphs

Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W with |B|=b and |W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to cographs, interval graphs, permutation graphs, distance-hereditary graphs, and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs.

Graph orientations with set connectivity requirements

In an undirected or a directed graph, the edge-connectivity between two disjoint vertex sets X and Y is defined as the minimum number of edges or arcs that should be removed for disconnecting all vertices in Y from those in X. This paper discusses how to construct a directed graph from a given undirected graph by orienting edges so as to preserve the edge-connectivity on pairs of vertex sets as much as possible. We present several bounds on the gap between the edge-connectivities in the undirected graph and in the obtained directed graphs, which extends the Nash-Williams’ strong orientation theorem.

The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

For a connected graph G = (V, E), an edge set $${S\subset E}$$ is called a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λ k (G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets $${U_1,U_2\subset V(G)}$$, denote the set of edges of G with one end in U 1 and the other in U 2 by [U 1, U 2]. Define $${\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U}$$ is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λ k -optimal if λk (G) = ξ k (G). A graph is said to be super-λk if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λk -optimal or super-λk . Moreover, we demonstrate that our results are best possible.

Optimal query complexity bounds for finding graphs

We consider the problem of finding an unknown graph by using queries with an additive property. This problem was partially motivated by DNA shotgun sequencing and linkage discovery problems of artificial intelligence. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask the sum of weights for weighted graphs. For a graph G with n vertices and at most m edges, we prove that there exists an algorithm to find the edges of G using O ( m log n 2 m log ( m + 1 ) ) queries of both types for all m. The bound is best possible up to a constant factor. For a weighted graph with a mild condition on weights, it is shown that O ( m log n log m ) queries are enough provided m ⩾ ( log n ) α for a sufficiently large constant α, which is best possible up to a constant factor if m ⩽ n 2 − ε for any constant ε > 0 . This settles, in particular, a conjecture of Grebinski [V. Grebinski, On the power of additive combinatorial search model, in: Proceedings of the 4th Annual International Conference on Computing and Combinatorics (COCOON 1998), Taipei, Taiwan, 1998, pp. 194–203] for finding an unweighted graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions as well as a similar coin weighing problem.