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Linear Time Algorithms for Hamiltonian Problems on (claw,net)Free Graphs
 SIAM J. COMPUTING
, 2000
"... We prove that clawfree graphs, containing an induced dominating path, have a Hamiltonian path, and that 2connected clawfree graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Ha ..."
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We prove that clawfree graphs, containing an induced dominating path, have a Hamiltonian path, and that 2connected clawfree graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)free graphs. These graphs enjoy those interesting structural properties.
Clawfree 3connected P11free graphs are hamiltonian
 COMPUTER SCIENCE, EMORY UNIVERSITY, ATLANTA
"... We show that every 3connected clawfree graph which contains no induced copy of P_11 is hamiltonian. Since there exist nonhamiltonian 3connected clawfree graphs without induced copies of P_12 this result is, in a way, best possible. ..."
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We show that every 3connected clawfree graph which contains no induced copy of P_11 is hamiltonian. Since there exist nonhamiltonian 3connected clawfree graphs without induced copies of P_12 this result is, in a way, best possible.
ClawFree and Generalized BullFree Graphs of Large Diameter Are Hamiltonian
, 1998
"... A generalized (i; j)bull B i;j is a graph obtained by identifying each of some two distinct vertices of a triangle with an endvertex of one of two vertexdisjoint paths of lengths i; j. We prove that every 2connected clawfree B 2;j free graph of diameter at least maxf7; 2jg (j 2) is hamilton ..."
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A generalized (i; j)bull B i;j is a graph obtained by identifying each of some two distinct vertices of a triangle with an endvertex of one of two vertexdisjoint paths of lengths i; j. We prove that every 2connected clawfree B 2;j free graph of diameter at least maxf7; 2jg (j 2) is hamiltonian.
Closure and forbidden pairs for 2factors
, 2010
"... Pairs of connected graphs X, Y such that a graph G being 2connected and XYfree implies G is hamiltonian were characterized by Bedrossian. Using the closure concept for clawfree graphs, the first author simplified the characterization by showing that if considering the closure of G, the list in th ..."
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Pairs of connected graphs X, Y such that a graph G being 2connected and XYfree implies G is hamiltonian were characterized by Bedrossian. Using the closure concept for clawfree graphs, the first author simplified the characterization by showing that if considering the closure of G, the list in the Bedrossian’s characterization can be reduced to one pair, namely, K1,3, N1,1,1 (where Ki,j is the complete bipartite graph, and Ni,j,k is the graph obtained by identifying endvertices of three disjoint paths of lengths i, j, k to the vertices of a triangle). Faudree et al. characterized pairs X, Y such that G being 2connected and XYfree implies G has a 2factor. Recently, the first author et al. strengthened the closure concept for clawfree graphs such that the closure of a graph has stronger properties while still preserving the (non)existence of a 2factor. In this paper we show that, using the 2factor closure, the list of forbidden pairs for 2factors can be reduced to two pairs, namely, K1,4, P4 and K1,3, N1,1,3. 1 Notation and terminology In this paper, by a graph we mean a simple finite undirected graph G = (V (G), E(G)), and for notations and terminology not defined here we refer to [3]. Specifically, Ck denotes the cycle on k vertices and Pk the path on k vertices (i.e. of length k − 1). A trivial path is a path having only one vertex, and a path with endvertices a, b is also referred to as an (a, b)path. For x ∈ V (G), dG(x) denotes the degree of x, and ∆(G) stands for the maximum degree of G, i.e. ∆(G) = max{dG(x)  x ∈ V (G)}. An edge e = uv ∈ E(G) is a pendant edge of G if min{dG(u), dG(v)} = 1 and max{dG(u), dG(v)} ≥ 3. The girth of a graph G, denoted g(G), is the length of a shortest cycle in G, and the circumference of G, denoted c(G), is the length of a longest cycle in G. A clique in a graph
Pancyclicity in Clawfree Graphs
"... In this paper, we present several conditions for K1,3free graphs, which guarantee the graph is subpancyclic. In particular, we show that every K1,3free graph with minimum degree sum δ2> 2 3n+ 1 − 4; every {K1,3, P7}free graph with δ2 ≥ 9; every {K1,3, Z4}free graph with δ2 ≥ 9; and every K1,3 ..."
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In this paper, we present several conditions for K1,3free graphs, which guarantee the graph is subpancyclic. In particular, we show that every K1,3free graph with minimum degree sum δ2> 2 3n+ 1 − 4; every {K1,3, P7}free graph with δ2 ≥ 9; every {K1,3, Z4}free graph with δ2 ≥ 9; and every K1,3free graph with maximum degree ∆, diam(G) < ∆+64 and δ2 ≥ 9 is subpancyclic. Key words: clawfree, pancyclicity, forbidden subgraphs 1
A Note on Hamiltonicity of Generalized NetFree Graphs of Large Diameter
"... A generalized (i; j; k)net N i;j;k is the graph obtained by identifying each of the vertices of a triangle with an endvertex of one of three vertexdisjoint paths of lengths i; j; k. We prove that every 2connected clawfree N 1;2;j free graph of diameter at least maxf7; 2jg (j 2) is hamiltonian ..."
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A generalized (i; j; k)net N i;j;k is the graph obtained by identifying each of the vertices of a triangle with an endvertex of one of three vertexdisjoint paths of lengths i; j; k. We prove that every 2connected clawfree N 1;2;j free graph of diameter at least maxf7; 2jg (j 2) is hamiltonian. Keywords: hamiltonian graphs, forbidden subgraphs, clawfree graphs 1991 Mathematics Subject Classification: 05C45 Research supported by grant GA CR No. 201/97/0407 1 1 Introduction In this paper we consider finite simple undirected graphs G = (V (G); E(G)) and for concepts and notations not defined here we refer the reader to [2]. For a set S ae V (G) we denote by N(S) the neighborhood of S, i.e. the set of all vertices of G which have a neighbor in S. If S = fxg, we simply write N(x) for N(fxg). For any subset M ae V (G), we denote NM (S) = N(S) " M . If H is a subgraph of G, we write NH (S) for N V (H) (S). For subsets M;N ae V (G), M " N = ;, we denote E(M;N) = fxy 2 E(G)j x...