On some balanced, totally balanced and submodular delivery games

This paper studies a class of delivery problems associated with the Chinese postman problem and a corresponding class of delivery games. A delivery problem in this class is determined by a connected graph, a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office. It is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server, who starts at the post office and is expected to traverse all edges in the graph before returning to the post office. A graph G is called Chinese postman-submodular, or, for short, CP-submodular (CP-totally balanced, CP-balanced, respectively) if for each delivery problem in which G is the underlying graph the associated delivery game is submodular (totally balanced, balanced, respectively). For undirected graphs we prove that CP-submodular graphs and CP-totally balanced graphs are weakly cyclic graphs and conversely. An undirected graph is shown to be CP-balanced if and only if it is a weakly Euler graph. For directed graphs, CP-submodular graphs can be characterized by directed weakly cyclic graphs. Further, it is proven that any strongly connected directed graph is CP-balanced. For mixed graphs it is shown that a graph is CP-submodular if and only if it is a mixed weakly cyclic graph. Finally, we note that undirected, directed and mixed weakly cyclic graphs can be recognized in linear time. Received May 20, 1997 / Revised version received August 18, 1998?Published online June 11, 1999     

On self-clique graphs with given clique sizes, II

The clique graph of a graph G is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques have a non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. We give a new characterization of the set of all connected self-clique graphs having all cliques but two of size 2.     

On self-intersections of immersed surfaces

A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.

     

Generalized Thrackle Drawings of Non-bipartite Graphs

A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.     

A Note on Lights-Out-Puzzle: Parity-State Graphs

A state of a graph G is an assignment of 0 or 1 to each vertex of G. A move of a state consists of choosing a vertex and then switching the value of the vertex as well as those of its neighbors. Two states are said to be equivalent if one state can be changed to the other by a series of moves. A parity-state graph is defined to be a graph in which two states are equivalent if and only if the numbers of 1’s in the two states have the same parity. We characterize parity-state graphs and present some constructions of parity-state graphs together with applications. Among other things, it is proved that the one-skeleton of the 3-polytope obtained from a simple 3-polytope by cutting off all vertices is a parity-state graph.     

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem [4] states that every planar triangle-free graph is 3-colorable. However, Voigt [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325-328] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangle-free graph without 7- and 8-cycles is 3-choosable.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

Netlike partial cubes, V: Completion and netlike classes

We define a completion of a netlike partial cube G by replacing each convex 2n-cycle C of G with n≥3 by an n-cube admitting C as an isometric cycle. We prove that a completion of G is a median graph if and only if G has the Median Cycle Property (MCP) (see N. Polat, Netlike partial cubes III. The Median Cycle Property, Discrete Math.). In fact any completion of a netlike partial cube having the MCP is defined by a universal property and turns out to be a minimal median graph containing G as an isometric subgraph. We show that the completions of the netlike partial cubes having the MCP preserves the principal constructions of these graphs, such as: netlike subgraphs, gated amalgams and expansions. Conversely any netlike partial cube having the MCP can be obtained from a median graph by deleting some particular maximal finite hypercubes. We also show that, given a netlike partial cube G having the MCP, the class of all netlike partial cubes having the MCP whose completions are isomorphic to those of G share different properties, such as: depth, lattice dimension, semicube graph and crossing graph.     

Switchings, extensions, and reductions in central digraphs

A directed graph is called central if its adjacency matrix A satisfies the equation A²=J, where J is the matrix with a 1 in each entry. It has been conjectured that every central directed graph can be obtained from a standard example by a sequence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We disprove these conjectures and present a general extension result which, in particular, shows that each counterexample extends to an infinite family.     

Every connected,locally connected nontrivial graph with no induced claw is hamiltonian

A graph is locally connected if every neighborthood induces a connected subgraph. We show here that every connected, locally connected graph on p ≥ 3 vertices and having no induced K_1,3 is Hamiltonian. Several sufficient conditions for a line graph to be Hamiltonian are obtained as corollaries.     

Deterministic vs non-deterministic graph property testing

A graph property \(\mathcal{P}\) is said to be testable if one can check whether a graph is close or far from satisfying \(\mathcal{P}\) using few random local inspections. Property \(\mathcal{P}\) is said to be non-deterministically testable if one can supply a “certificate” to the fact that a graph satisfies \(\mathcal{P}\) so that once the certificate is given its correctness can be tested. The notion of non-deterministic testing of graph properties was recently introduced by Lovász and Vesztergombi [9], who proved that (somewhat surprisingly) a graph property is testable if and only if it is non-deterministically testable. Their proof used graph limits, and so it did not supply any explicit bounds. They thus asked if one can obtain a proof of their result which will supply such bounds. We answer their question positively by proving their result using Szemerédi’s regularity lemma. An interesting aspect of our proof is that it highlights the fact that the regularity lemma can be interpreted as saying that all graphs can be approximated by finitely many “template” graphs.     

Some topics in cochromatic theory

GivenG, a graph, the cochromatic number,Z(G), ofG is the fewest number of sets into which the vertex set can be partitioned so that each set induces a complete or an empty graph. A graph is critically cochromatic if the removal of any of its vertices decreases its cochromatic number. A graph is uniquely cochromatic if there is exactly one partition of minimum order in which each set induces a complete or an empty graph. A graph is comaximal if the removal of any edge increases its cochromatic number. These and related concepts are examined.     

Recognizing String Graphs Is Decidable

Abstract. A graph is called a string graph if its vertices can be represented by continuous curves (``strings') in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times. We obtain as a corollary that there is an algorithm for deciding whether a given graph is a string graph. This solves an old problem of Benzer (1959), Sinden (1966), and Graham (1971).     

Relaxations of vertex packing

A polynomially computable upper bound for the weighted independence number of a graph is studied. This gives rise to a convex body containing the vertex packing polytope of the graph. This body is a polytope if and only if the graph is perfect. As an application of these notions, we show that the maximum weight independent set of an h-perfect graph can be found in polynomial time.     

Fall colouring of bipartite graphs and cartesian products of graphs

The question of whether a graph can be partitioned into k independent dominating sets, which is the same as having a fallk-colouring, is considered. For k=3, it is shown that a graph G can be partitioned into three independent dominating sets if and only if the cartesian product G□K₂ can be partitioned into three independent dominating sets. The graph K₂ can be replaced by any graph H such that there is a mapping f:Q_n→H, where f is a type-II graph homomorphism.The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of k, iterated cartesian products are considered, leading to a result that shows for what values of k the hypercubes can be partitioned into k independent dominating sets.     

Visibility Graphs and Oriented Matroids

Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Multicyclic treelike reflexive graphs

A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles.     

Perfect k-line graphs and k-total graphs

The concept of the line graph can be generalized as follows. The k-line graph L_k(G) of a graph G is defined as a graph whose vertices are the complete subgraphs on k vertices in G. Two distinct such complete subgraphs are adjacent in L_k(G) if and only if they have in G k ? 1 vertices in common. The concept of the total graph can be generalized similarly. Then the Perfect Graph Conjecture will be proved for 3-line graphs and 3-total graphs. Moreover, perfect 3-line graphs are not contained in any of the known classes of perfect graphs. © 1993 John Wiley & Sons, Inc.     

Visibility Graphs and Oriented Matroids

   Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Contracting planar graphs to contractions of triangulations

For every graph H, there exists a polynomial-time algorithm deciding if a planar input graph G can be contracted to H. However, the degree of the polynomial depends on the size of H. We identify a class of graphs C such that for every fixed H∈C, there exists a linear-time algorithm deciding whether a given planar graph G can be contracted to H. The class C is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph H∈C if and only if there exists a constant cH such that if the treewidth of a graph is at least cH, it contains H as a contraction. We also provide a characterization of C in terms of minimal forbidden contractions.