Recognizing Planar Strict Quasi-Parity Graphs

 A graph is a strict-quasi parity (SQP) graph if every induced subgraph that is not a clique contains a pair of vertices with no odd chordless path between them (an “even pair”). We present an O(n ³) algorithm for recognizing planar strict quasi-parity graphs, based on Wen-Lian Hsu's decomposition of planar (perfect) graphs and on the (non-algorithmic) characterization of planar minimal non-SQP graphs given in [9]. Received: September 21, 1998 Final version received: May 9, 2000     

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.Research supported in part by Office of Naval Research Contract N00014-88-K-0018.Research supported in part by Office of Naval Research Contract N00014-88-K-0163.     

A Ramsey-type theorem for traceable graphs

A graph G is traceable if there is a path passing through all the vertices of G. It is proved that every infinite traceable graph either contains arbitrarily large finite chordless paths, or contains a subgraph isomorphic to graph A, illustrated in the text. A corollary is that every finitely generated infinite lattice of length 3 contains arbitrarily large finite fences. It is also proved that every infinite traceable graph containing no chordless four-point path contains a subgraph isomorphic to K_ω,ω. The versions of these results for finite graphs are discussed.     

Longest cycles in 3-connected graphs contain three contractible edges

We show that if G is a 3-connected graph of order at least seven, then every longest path between distinct vertices in G contains at least two contractible edges. An immediate corollary is that longest cycles in such graphs contain at least three contractible edges.     

Graphoidal bipartite graphs

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths inG such that every path in ψ has at least two vertices, every vertex ofG is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. Let Ω (ψ) denote the intersection graph of ψ. A graph G is said to be graphoidal if there exists a graphH and a graphoidal cover ψof H such that G is isomorphic to Ω(ψ). In this paper we study the properties of graphoidal graphs and obtain a forbidden subgraph characterisation of bipartite graphoidal graphs.     

A characterization of P4‐indifference graphs

A graph is a P₄‐indifference graph if it admits a linear ordering ≺ on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has either a ≺ b ≺ c ≺ d or d ≺ c ≺ b ≺ a. P₄‐indifference graphs generalize indifference graphs and are perfectly orderable. We give a characterization of P₄‐indifference graphs by forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 155‐162, 1999     

A proof of Hougardy's conjecture for diamond-free graphs

A pair of vertices of a graph is called an even pair if every chordless path between them has an even number of edges. A graph is minimally even pair free if it is not a clique, contains no even pair, but every proper induced subgraph either contains an even pair or is a clique. Hougardy (European J. Combin. 16 (1995) 17–21) conjectured that a minimally even pair free graph is either an odd cycle of length at least five, the complement of an even or odd cycle of length at least five, or the linegraph of a bipartite graph. A diamond is a graph obtained from a complete graph on four vertices by removing an edge. In this paper we verify Hougardy's conjecture for diamond-free graphs by adapting the characterization of perfect diamond-free graphs given by Fonlupt and Zemirline (Maghreb Math. Rev. 1 (1992) 167–202).     

Fragile graphs with small independent cuts

A graph is called fragile if it has a vertex cut which is also an independent set. Chen and Yu proved that every graph with n vertices and at most 2n?4 edges is fragile, which was conjectured to be true by Caro. However, their proof does not give any information on the number of vertices in the independent cuts. The purpose of this paper is to investigate when a graph has a small independent cut. We show that if G is a graph on n vertices and at most (12n/7)?3 edges, then G contains an independent cut S with ∣S∣≤3. Upper bounds on the number of edges of a graph having an independent cut of size 1 or 2 are also obtained. We also show that for any positive integer k, there is a positive number ε such that there are infinitely many graphs G with n vertices and at most (2?ε)n edges, but G has no independent cut with less than k vertices. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 327–341, 2002     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

RandomlyH-coverable graphs

A graph israndomly matchable if every matching of the graph is contained in a perfect matching. We generalize this notion and say that a graphG israndomly H-coverable if every set of independent subgraphs, each isomorphic toH, that does not cover the vertices ofG can be extended to a larger set of independent copies ofH. Various problems are considered for the situation whereH is a path. In particular, we characterize the graphs that are randomlyP ₃ -coverable.     

On brittle graphs

Chvátal defined a graph G to be brittle if each induced subgraph F of G contains a vertex that is not a midpoint of any P₄ or not an endpoint of any P₄. Every brittle graph is perfectly orderable. In this paper, we prove that a graph is brittle whenever it is HHD-free (containing no chordless cycle with at least five vertices, no cycle on six vertices with a long chord, and no complement of the chordless path on five vertices). We also design an O(n₄) algorithm to recognize HHD-free graphs, and also an O(n₄) algorithm to construct a perfect order of an HHD-free graph. It follows from this result that an optimal coloring and a largest clique of an HHD-free graph can be found in O(n₄) time.     

Bidegreed graphs are edge reconstructible

An edge-deleted subgraph of a graph G is a subgraph obtained from G by the deletion of an edge. The Edge Reconstruction Conjecture asserts that every simple finite graph with four or more edges is determined uniquely, up to isomorphism, by its collection of edge-deleted subgraphs. A class of graphs is said to be edge reconstructible if there is no graph in the class with four or more edges that is not edge reconstructible. This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible. The results are then generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructible.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

On the P4-structure of perfect graphs. II. Odd decompositions

We prove that a graph is perfect if its vertices can be coloured by two colours in such a way that each induced chordless path with four vertices has an odd number of vertices of each colour. Using this result, we prove a decomposition theorem for perfect graphs; this theorem is defined in terms of the chordless path with four vertices.     

Generating weakly triangulated graphs

We show that a graph is weakly triangulated, or weakly chordal, if and only if it can be generated by starting with a graph with no edges, and repeatedly adding an edge, so that the new edge is not the middle edge of any chordless path with four vertices. This is a corollary of results due to Sritharan and Spinrad, and Hayward, Hoång and Maffray, and a natural analog of a theorem due to Fulkerson and Gross, which states that a graph is triangulated, or chordal, if and only if it can be generated by starting with a graph with no vertices, and repeatedly adding a vertex, so that the new vertex is not the middle vertex of any chordless path with three vertices. Our result answers the question of whether there exists a composition scheme that generates exactly the class of weakly triangulated graphs. © 1996 John Wiley & Sons, Inc.     

On the structure of bull-free perfect graphs

A bull is a graph obtained by adding a pendant vertex at two vertices of a triangle. Chvátal and Sbihi showed that the Strong Perfect Graph Conjecture holds for bull-free graphs. We show that bull-free perfect graphs are quasi-parity graphs, and that bull-free perfect graphs with no antihole are perfectly contractile. Our proof yields a polynomial algorithm for coloring bull-free strict quasi-parity graphsPartially supported by CNPq, grant 30 1160/91.0     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

Paths and edge-connectivity in graphs

Mader proved that for every k-edge-connected graph G (k ≥ 4), there exists a path joining two given vertices such that the subgraph obtained from G by deleting the edges of the path is (k - 2)-edge-connected. A generalization of this and a sufficient condition for existance of 3, 4, or 5 terminus k edge-disjoint paths in graphs are given.     

On the two-edge-colorings of perfect graphs

We investigate the conjecture that a graph is perfect if it admits a two-edge-coloring such that two edges receive different colors if they are the nonincident edges of a P₄ (chordless path with four vertices). Partial results on this conjecture are given in this paper. © 1995 John Wiley & Sons, Inc.     

Preperfect graphs

We say that a vertexx of a graph is predominant if there exists another vertexy ofG such that either every maximum clique ofG containingy containsx or every maximum stable set containingx containsy. A graph is then called preperfect if every induced subgraph has a predominant vertex. We show that preperfect graphs are perfect, and that several well-known classes of perfect graphs are preperfect. We also derive a new characterization of perfect graphs.