Showing papers in "Journal of Graph Theory in 2008"
TL;DR: In this paper, the authors adapted the δ-hyperbolic concept to very large but finite graphs and used the Cartan-Alexandrov-Toponogov (CAT) theory to derive the thresholding value of δdiam below which the geometry has negative curvature properties.
Abstract: In this article, the δ-hyperbolic concept, originally developed for infinite graphs, is adapted to very large but finite graphs. Such graphs can indeed exhibit properties typical of negatively curved spaces, yet the traditional δ-hyperbolic concept, which requires existence of an upper bound on the fatness δ of the geodesic triangles, is unable to capture those properties, as any finite graph has finite δ. Here the idea is to scale δ relative to the diameter of the geodesic triangles and use the Cartan–Alexandrov–Toponogov (CAT) theory to derive the thresholding value of δdiam below which the geometry has negative curvature properties. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 157–180, 2008
94 citations
TL;DR: In this paper, it was shown that for any hypergraph of order n and size m with edge sizes at least 3, a transversal of size at most (n + m)-4 has at most one half of its order.
Abstract: It is shown in several manuscripts [Discrete Math 86 (1990), 117–126; Combinatorica 12 (1992), 19–26] that every hypergraph of order n and size m with edge sizes at least 3 has a transversal of size at most (n + m)-4. In this article, we characterize the connected such hypergraphs that achieve equality in this bound. As a direct consequence of this bound, the total domination of a graph with minimum degree at least 3 is at most one-half its order. An elegant graph theoretic proof of this result is presented in Archdeacon et al. [J Graph Theory 46 (2004), 207–210]. Using our hypergraph result, we characterize the connected graphs with minimum degree at least 3 and with total domination number exactly one-half their order. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 326–348, 2008
70 citations
TL;DR: In this article, it was shown that if G is a planar graph with Δ(G) = 3 and girth G = 9, then χl(G2) ≈ 6.
Abstract: The squareG2 of a graph G is the graph with the same vertex set G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that every planar graph G with maximum degree Δ(G) = 3 satisfies χ(G2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G2 equals the chromatic number of G2, that is, χl(G2) = χ(G2) for all G. If true, this conjecture (together with Thomassen's result) implies that every planar graph G with Δ(G) = 3 satisfies χl(G2) ≤ 7. We prove that every connected graph (not necessarily planar) with Δ(G) = 3 other than the Petersen graph satisfies χl(G2) ≤8 (and this is best possible). In addition, we show that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 7, then χl(G2) ≤ 7. Dvořak, Skrekovski, and Tancer showed that if G is a planar graph with Δ(G) = 3 and girth g(G) ≥ 10, then χl(G2) ≤6. We improve the girth bound to show that if G is a planar graph with Δ(G) = 3 and g(G) ≥ 9, then χl(G2) ≤ 6. All of our proofs can be easily translated into linear-time coloring algorithms. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 65–87, 2008
53 citations
TL;DR: In this article, it was shown that rank-width of a graph is less than or equal to the branchwidth of the graph unless the branch-width is 0 or 1.
Abstract: We prove that the rank-width of the incidence graph of a graph G is either equal to or exactly one less than the branch-width of G, unless the maximum degree of G is 0 or 1. This implies that rank-width of a graph is less than or equal to branch-width of the graph unless the branch-width is 0. Moreover, this inequality is tight. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 239–244, 2008
52 citations
TL;DR: In this article, King et al. showed that the conjecture holds if G is a quasi-line graph, extending a result of King et. al. who proved the conjecture for line graphs.
Abstract: A quasi-line graph is a graph in which the neighborhood of any vertex can be covered by two cliques; every line graph is a quasi-line graph. Reed conjectured that for any graph G, $\chi({{G}}) \leq\left \lceil {{{1}}\over {{2}}}(\Delta({{G}})+{{1}}+\omega({{G}}))\right\rceil$ [Reed, J Graph Theory 27 (1998), 177–212]. We prove that the conjecture holds if G is a quasi-line graph, extending a result of King et al. who proved the conjecture for line graphs [Eur J Comb 28 (2007), 2182–2187], and improving the bound of $\chi{{(}}{{G}}{{)}} \leq {3\over 2} \omega({{G}})$ given by Chudnovsky and Ovetsky [J Graph Theory 54 (2007), 41–50]. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 215–228, 2008
50 citations
TL;DR: In this article, an Erdos-type result on the k-domination number of a graph G is given, where every vertex not in S has at least k neighbors in S. The upper bound on the minimum cardinality of a k dominating set of G is known in terms of the order n and the minimum degree of G.
Abstract: A subset S of vertices of a graph G is k-dominating if every vertex not in S has at least k neighbors in S. The k-domination number $\gamma_k(G)$ is the minimum cardinality of a k-dominating set of G. Different upper bounds on $\gamma_{k}(G)$ are known in terms of the order n and the minimum degree $\delta$ of G. In this self-contained article, we present an Erdos-type result, from which some of these bounds follow. In particular, we improve the bound $\gamma_{k}(G) \le (2k- \delta - 1)n/(2k -\delta)$ for $(\delta +3)/2 \le k \le \delta - 1$, proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality $\gamma_{k}(G) \le kn/(k +1)$, if $k \le \delta$, of Cockayne et al. This characterization generalizes that of graphs realizing $\gamma_1(G) = \gamma(G) = n/2$. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 33–40, 2008
50 citations
TL;DR: Forb(n, H) as discussed by the authors is the set of all graphs on n vertices that have no induced copy of H. In this paper, we use the binary chromatic number of the graph H to measure the editing distance from H to H.
Abstract: An edge-operation on a graph G is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs $\cal G$, the editing distance from G to $\cal G$ is the smallest number of edge-operations needed to modify G into a graph from $\cal G$. In this article, we fix a graph H and consider Forb(n, H), the set of all graphs on n vertices that have no induced copy of H. We provide bounds for the maximum over all n-vertex graphs G of the editing distance from G to Forb(n, H), using an invariant we call the binary chromatic number of the graph H. We give asymptotically tight bounds for that distance when H is self-complementary and exact results for several small graphs H. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:123–138, 2008
46 citations
TL;DR: In this article, the pebbling number ΠOPT(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r.
Abstract: Given a distribution of pebbles on the vertices of a graph G, a pebbling move takes two pebbles from one vertex and puts one on a neighboring vertex. The pebbling number Π(G) is the least k such that for every distribution of k pebbles and every vertex r, a pebble can be moved to r. The optimal pebbling number ΠOPT(G) is the least k such that some distribution of k pebbles permits reaching each vertex.
Using new tools (such as the “Squishing” and “Smoothing” Lemmas), we give short proofs of prior results on these parameters for paths, cycles, trees, and hypercubes, a new linear-time algorithm for computing Π(G) on trees, and new results on ΠOPT(G). If G is connected and has n vertices, then $\Pi_{\rm OPT}(G)\le \lceil{2n/3}\rceil$ (sharp for paths and cycles). Let an,k be the maximum of ΠOPT(G) when G is a connected n-vertex graph with δ(G) ≥ k. Always $2 \lceil{n \over {k+1}}\rceil \le a_{{n},k}\le 4 \lceil{n \over {k+1}}\rceil$, with a better lower bound when k is a nontrivial multiple of 3. Better upper bounds hold for n-vertex graphs with minimum degree k having large girth; a special case is $\Pi_{{\rm OPT}}({G})\le {16}{n}/{({k}^{2}+{17})}$ when G has girth at least 5 and k ≥ 4. Finally, we compute ΠOPT(G) in special families such as prisms and Mobius ladders. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 215–238, 2008
46 citations
TL;DR: In this article, the authors give a characterization of the trees with given maximum degree which maximize the number of independent subsets, and show that these trees also minimize the independent edge subsets.
Abstract: The number of independent vertex subsets is a graph parameter that is, apart from its purely mathematical importance, of interest in mathematical chemistry. In particular, the problem of maximizing or minimizing the number of independent vertex subsets within a given class of graphs has already been investigated by many authors. In view of the applications of this graph parameter, trees of restricted degree are of particular interest. In the current article, we give a characterization of the trees with given maximum degree which maximize the number of independent subsets, and show that these trees also minimize the number of independent edge subsets. The structure of these trees is quite interesting and unexpected: it can be described by means of a novel digital system—in the case of maximum degree 3, we obtain a binary system using the digits 1 and 4. The proof mainly depends on an exchange lemma for branches of a tree. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 49–68, 2008
Dedicated to Prof. Robert Tichy on the occasion of his 50th birthday.
This article was written while C. Heuberger was a visitor at the Center of Experimental Mathematics at the University of Stellenbosch. He thanks the center for its hospitality.
43 citations
TL;DR: The linear arboricity of a graph G is defined as the minimum number of linear forests which partition the edges of G as discussed by the authors, and the conjecture is also true for planar graphs.
Abstract: The linear arboricity of a graph G is the minimum number of linear forests which partition the edges of G. Akiyama et al. conjectured that $\lceil {\Delta {({G})}\over {2}}\rceil \leq {la}({G}) \leq \lceil {\Delta({G})+{1}\over {2}}\rceil$ for any simple graph G. Wu wu proved the conjecture for a planar graph G of maximum degree $\Delta
ot={{7}}$. It is noted here that the conjecture is also true for $\Delta={{7}}$. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:210-220, 2008
40 citations
TL;DR: In this article, it was shown that a 171-edge-connected graph has an edge decomposition into paths of length 3 if and only its size is divisible by 3.
Abstract: We prove that a 171-edge-connected graph has an edge-decomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, and whether 3-edge-connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008
TL;DR: In this article, the best upper bound of Frieze et al. was improved to 48 (n-δ) + 6 by a log n factor for d-regular graphs with minimum degree δ ≥ 104-3 n2-3 log 1-4 n.
Abstract: Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f: E(G) → {1, 2,…, w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f(v) are distinct. The smallest w for which there exists an irregular assignment on G is called the irregularity strength of G, and it is denoted by s(G). We show that if the minimum degree δ (G) ≥ 10n3-4 log1-4 n, then s(G) ≤ 48 (n-δ) + 6. For these δ, this improves the magnitude of the previous best upper bound of Frieze et al. by a log n factor. It also provides an affirmative answer to a question of Lehel, whether for every ± ∈ (0, 1), there exists a constant c = c( α) such that s(G) ≤ c for every graph G of order n with minimum degree δ(G) ≥ (1 - α)n. Specializing the argument for d-regular graphs with d ≥ 104-3 n2-3 log1-3 n, we prove that s(G) ≤ 48 (n-d) + 6. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 299313, 2008
TL;DR: It is proved that the cop number of any n-vertex graph G, denoted by G, is at most at most ${{O}}\big({{{n}}\over {{\rm lg}} {{n}}}\big) .
Abstract: In this note, we prove that the cop number of any n-vertex graph G, denoted by ${{c}}({{G}})$, is at most ${{O}}\big({{{n}}\over {{\rm lg}} {{n}}}\big)$. Meyniel conjectured ${{c}}({{G}})={{O}}(\sqrt{{{n}}})$. It appears that the best previously known sublinear upper-bound is due to Frankl, who proved ${{c}}({{G}})\leq ({{1}}+ {{o}}({{1}})){{{n}}{{\rm lg}}{{\rm lg}} {{n}}\over {{\rm lg}} {{n}}}$. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 45–48, 2008
TL;DR: In this article, it was shown that for any multigraph G = (V, E) which has no loops and is of order n, size m, and maximum degree Δ < 10-3m/√8n, there is a mapping f:V∪E➝{1,2,…,[m+2/3]} such that f(u)+f(uv)+f (v)≠ f (u')+f(u'v')+ f(v') for every uv, u'v'eE
Abstract: As our main result, we prove that for every multigraph G = (V, E) which has no loops and is of order n, size m, and maximum degree Δ < 10-3m/√8n there is a mapping f:V∪E➝{1,2,…,[m+2/3]} such that f(u)+f(uv)+f(v)≠ f(u')+f(u'v')+f(v') for every uv,u'v'eE with uv≠u'v'. Functions with this property were recently introduced and studied by Baca et al. and were called edge irregular total labelings. Our result confirms a recent conjecture of Ivanco and Jendrol2 about such labelings for dense graphs, for graphs where the maximum and minimum degree are not too different in terms of the order, and also for large graphs of bounded maximum degree. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 333343, 2008
TL;DR: In this paper, the fundamental cycles of a spanning tree are shown to generate the cycle space in a connected, compact, weakly Hausdorff edge space, and in such a space, the orthogonal complement of the bond space is the cycle.
Abstract: We develop a general model of edge spaces in order to generalize, unify, and simplify previous work on cycle spaces of infinite graphs. We give simple topological criteria to show that the fundamental cycles of a (generalization of a) spanning tree generate the cycle space in a connected, compact, weakly Hausdorff edge space. Furthermore, in such a space, the orthogonal complement of the bond space is the cycle space. This work unifies the two different notions of cycle space as introduced by Diestel and Kuhn [Combinatorica 24 (2004), 68–89 and Eur J Combin 25 (2004), 835–862] and by Bonnington and Richter [J Graph Theory 44 (2003), 132–147]. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 115–144, 2008
TL;DR: In this article, it was shown that the Cartesian product of two forests has game chromatic number at most 10, and two planar graphs with game coloring at most 105.
Abstract: This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product GsG′ has game chromatic number at most k(k + m - 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008
TL;DR: It is improved upon this bound, showing that it is sufficient to take n ≥ 2k - 1 ( k - 1) - 1 to guarantee that in any two-coloring of the edges of the complete graph Kn,n there is a monochromatic Kk,k.
Abstract: We consider the following question: how large does n have to be to guarantee that in any two-coloring of the edges of the complete graph Kn,n there is a monochromatic Kk,k? In the late 1970s, Irving showed that it was sufficient, for k large, that n ≥ 2k - 1 (k - 1) - 1. Here we improve upon this bound, showing that it is sufficient to take $${n} \geq ({1} + {o}({1})) {2}^{{k}+ {1}}\; {\log}\; {k},$$ where the log is taken to the base 2. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 351356, 2008
TL;DR: In this article, the authors characterize homomorphism-homogeneous finite tournaments where vertices are allowed to have loops, and show that a structure is homomorphic if every isomorphism between finite substructures extends to an automorphism of the structure.
Abstract: A structure is called homogeneous if every isomorphism between finite substructures of the structure extends to an automorphism of the structure. Recently, Cameron and Nesetril introduced a relaxed version of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this article we characterize homomorphism-homogeneous finite tournaments where vertices are allowed to have loops. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 45–58, 2008
TL;DR: In this article, it was shown that a graph on p vertices with q edges has at most (1 + ) 2.5 cycles, and if G is planar, then it has 2.
Abstract: Let G be a graph on p vertices with q edges and let r = q - p = 1. We show that G has at most ${15\over 16} 2^{r}$ cycles. We also show that if G is planar, then G has at most 2r - 1 = o(2r - 1) cycles. The planar result is best possible in the sense that any prism, that is, the Cartesian product of a cycle and a path with one edge, has more than 2r - 1 cycles. © Wiley Periodicals, Inc. J. Graph Theory 57: 255–264, 2008
TL;DR: In this paper, a finite graph G is partitioned into classes of size α d, where α≥ 11-4 + ε+ε + δ, and each class receives all αd colors.
Abstract: Let η > 0 be given. Then there exists d0 = d0(η) such that the following holds. Let G be a finite graph with maximum degree at most d ≥ d0 whose vertex set is partitioned into classes of size α d, where α≥ 11-4 + η. Then there exists a proper coloring of G with αd colors in which each class receives all αd colors. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:148–158, 2008
TL;DR: A graph G is a quasi-line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques.
Abstract: A graph G is a quasi-line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi-line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t-colorable then it contains Kt + 1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi-line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008
This research was conducted while the author served as a Clay Mathematics Institute Research Fellow. Part of the research was conducted at Princeton University.
TL;DR: For a graph G with automorphisms, ν vertices, and E edges, it is natural to define the Ramsey multiplicity constant C(G) to be the minimum number of monochromatic copies of G over all 2-colorings of the edges of the complete graph Kn as mentioned in this paper.
Abstract: The Ramsey multiplicity M(G;n) of a graph G is the minimum number of monochromatic copies of G over all 2-colorings of the edges of the complete graph Kn. For a graph G with a automorphisms, ν vertices, and E edges, it is natural to define the Ramsey multiplicity constant C(G) to be $\lim_{{n} \to \infty} {M}(G;n){a/v}!{n \choose v}$, which is the limit of the fraction of the total number of copies of G which must be monochromatic in a 2-coloring of the edges of Kn. In 1980, Burr and Rosta showed that 0 ≥ C(G) ≤ 21-E for all graphs G, and conjectured that this upper bound is tight. Counterexamples of Burr and Rosta's conjecture were first found by Sidorenko and Thomason independently. Later, Clark proved that there are graphs G with E edges and 2E-1C(G) arbitrarily small. We prove that for each positive integer E, there is a graph G with E edges and C(G) ≤ E-E/2 + o(E). © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 89–98, 2008
TL;DR: In this paper, the bipartite neighborhood recognition problem was shown to be polynomial-time solvable, or equivalent with graph isomorphism, and the uniqueness of neighborhood realizations of hypergraphs was studied.
Abstract: Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP-hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP-hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial-time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008
TL;DR: A plane graph G is coupled k-choosable if, for any list assignment L satisfying L satisfying $|L| = k| for every vertex, there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors as discussed by the authors.
Abstract: A plane graph G is coupled k-choosable if, for any list assignment L satisfying $|{{L}}({{x}})|= {{k}}$ for every ${{x}}\in {{V}}({{G}})\cup {{F}}({{G}})$, there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7-choosable. We further show that maximal plane graphs, ${{K}}_{{4}}$-minor free graphs, and plane graphs with maximum degree at most three are coupled 6-choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008
TL;DR: In this paper, the existence of a (g, f)-factor in edge-weighted graphs was shown to be a necessary and sufficient condition for any edgeweighted graph.
Abstract: A (g, f)-factor of a graph is a subset F of E such that for all $v \in V$, $g(v)\le {\rm deg}_{F}(v)\le f(v)$. Lovasz gave a necessary and sufficient condition for the existence of a (g, f)-factor. We extend, to the case of edge-weighted graphs, a result of Kano and Saito who showed that if $g(v)< \lambda {\rm deg}_{E}(v) < f (v)$ for any $\lambda\in [0,1]$, then a (g, f)-factor always exist. In addition, we use results of Anstee to provide new necessary and sufficient conditions for the existence of a (g, f)-factor. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 265–274, 2008
Abstract: We show that a typical d-regular graph G of order n does not contain an induced forest with around 2Ind/d vertices, when n >> d >> 1, this bound being best possible because of a result of Frieze and Luczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149156, 2008
TL;DR: In this paper, it was shown that every cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length, such that the graph can be represented by a straight-line segment.
Abstract: We prove that every simple cubic planar graph admits a planar embedding such that each edge is embedded as a straight line segment of integer length. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:270-274, 2008
TL;DR: In this article, it was shown that every simple graph with at least one edge contains a non-residual edge, which was conjectured by El-Zahar in 1997.
Abstract: An induced subgraph ${\cal S}$ of a graph ${\cal G}$ is called a derived subgraph of ${\cal G}$ if ${\cal S}$ contains no isolated vertices. An edge e of ${\cal S}$ is said to be residual if e occurs in more than half of the derived subgraphs of ${\cal S}$. In this article, we prove that every simple graph with at least one edge contains a non-residual edge. This was conjectured by El-Zahar in 1997. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 344–352, 2008
TL;DR: Bondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC) as mentioned in this paper, and this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well.
Abstract: Bondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC). We show that this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well. Specifically, if G does not have an isolated vertex then G s P2 and G s C2k have SCDCs. If G has an SCDC then so does G s Pk, k > 2 and G s C2k + 1. We use these Cartesian results to show that P2j[G] (j ≥ 1) and Ck[G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P2j + 1[G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of Kn,n into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008
TL;DR: In this article, the degree-sequence-forcing sets are defined for graphs with cardinality at most two, and a complete characterization of the degree sequence forcing sets is given when the cardinality is at most 2.
Abstract: Given a set ${\cal F}$ of graphs, a graph G is ${\cal F}$-free if G does not contain any member of ${\cal F}$ as an induced subgraph. We say that ${\cal F}$ is a degree-sequence-forcing set if, for each graph G in the class ${\cal C}$ of ${\cal F}$-free graphs, every realization of the degree sequence of G is also in ${\cal C}$. We give a complete characterization of the degree-sequence-forcing sets ${\cal F}$ when ${\cal F}$ has cardinality at most two. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 131–148, 2008