11 research outputs found
Parameterized Edge Hamiltonicity
We study the parameterized complexity of the classical Edge Hamiltonian Path
problem and give several fixed-parameter tractability results. First, we settle
an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT
parameterized by vertex cover, and that it also admits a cubic kernel. We then
show fixed-parameter tractability even for a generalization of the problem to
arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set.
We also consider the problem parameterized by treewidth or clique-width.
Surprisingly, we show that the problem is FPT for both of these standard
parameters, in contrast to its vertex version, which is W-hard for
clique-width. Our technique, which may be of independent interest, relies on a
structural characterization of clique-width in terms of treewidth and complete
bipartite subgraphs due to Gurski and Wanke
On Eulerian subgraphs and hamiltonian line graphs
A graph {\color{black}} is Hamilton-connected if for any pair of distinct vertices {\color{black}}, {\color{black}} has a spanning -path; {\color{black}} is 1-hamiltonian if for any vertex subset with , has a spanning cycle. Let , and denote the minimum degree, the matching number and the line graph of a graph , respectively. The following result is obtained. {\color{black} Let be a simple graph} with . If , then each of the following holds. \\ (i) is Hamilton-connected if and only if . \\ (ii) is 1-hamiltonian if and only if . %==========sp For a graph , an integer and distinct vertices , an -path-system of is a subgraph consisting of internally disjoint -paths. The spanning connectivity is the largest integer such that for any with and for any with , has a spanning -path-system. It is known that , and determining if is an NP-complete problem. A graph is maximally spanning connected if . Let and be the smallest integers and such that is maximally spanning connected and , respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for and , and characterized the extremal graphs reaching the upper bounds. %==============st For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph on vertices with , when is sufficiently large, is -supereulerian or is contractible to . We prove the following for any nonnegative integers and . \\ (i) For any real numbers and with , there exists a family of finitely many graphs \F(a,b;s,t) such that if is a simple graph on vertices with and , then either is -supereulerian, or is contractible to a member in \F(a,b;s,t). \\ (ii) Let denote the connected loopless graph with two vertices and parallel edges. If is a simple graph on vertices with and , then when is sufficiently large, either is -supereulerian, or for some integer with , is contractible to a . %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and , and proposed a few problems to determine \cp(a,b) with when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let denote the essential edge-connectivity of a graph , and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and . We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of , \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
Circuits and Cycles in Graphs and Matroids
This dissertation mainly focuses on characterizing cycles and circuits in graphs, line graphs and matroids. We obtain the following advances.
1. Results in graphs and line graphs. For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent (s, t)-path. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following. (i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected. (ii) pc(G) ≤ l(G) + 2. Furthermore, if l(G) ≥ 2, then pc(G) = l(G) + 2 if and only if for some integer t ≥ 3, G has a bridge divalent (3, t)-path of length l(G).
For a graph G, the supereulerian width μ′(G) of a graph G is the largest integer s such
that G has a spanning (k;u,v)-trail-system, for any integer k with 1 ≤ k ≤ s, and for any
u, v ∈ V (G) with u ̸= v. Thus μ′(G) ≥ 2 implies that G is supereulerian, and so graphs with
higher supereulerian width are natural generalizations of supereulerian graphs. Settling an open
problem of Bauer, Catlin in [J. Graph Theory 12 (1988), 29-45] proved that if a simple graph
G on n ≥ 17 vertices satisfy δ(G) ≥ n − 1, then μ′(G) ≥ 2. In this paper, we show that for 4
any real numbers a, b with 0 \u3c a \u3c 1 and any integer s \u3e 0, there exists a finite graph family
F = F(a,b,s) such that for a simple graph G with n = |V(G)|, if for any u,v ∈ V(G) with
uv ∈/ E(G), max{dG(u), dG(v)} ≥ an + b, then either μ′(G) ≥ s + 1 or G is contractible to a
member in F. When a = 1,b = −3, we show that if n is sufficiently large, K3,3 is the only 42
obstacle for a 3-edge-connected graph G to satisfy μ′(G) ≥ 3. An hourglass is a graph obtained from K5 by deleting the edges in a cycle of length 4, and an
hourglass-free graph is one that has no induced subgraph isomorphic to an hourglass. Kriesell in [J. Combin. Theory Ser. B, 82 (2001), 306-315] proved that every 4-connected hourglass-free line graph is Hamilton-connected, and Kaiser, Ryj ́aˇcek and Vr ́ana in [Discrete Mathematics, 321 (2014) 1-11] extended it by showing that every 4-connected hourglass-free line graph is 1- Hamilton-connected. We characterize all essentially 4-edge-connected graphs whose line graph is hourglass-free. Consequently we prove that for any integer s and for any hourglass-free line
graph L(G), each of the following holds. (i) If s ≥ 2, then L(G) is s-hamiltonian if and only if κ(L(G)) ≥ s + 2; (ii) If s ≥ 1, then L(G) is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3.
For integers s1, s2, s3 \u3e 0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1,s2 and s3, respectively. We prove the following results. (i)LetN1 ={Ns1,s2,s3 :s1 \u3e0,s1 ≥s2 ≥s3 ≥0ands1+s2+s3 ≤6}. Thenforany N ∈ N1, every N-free line graph L(G) with |V (L(G))| ≥ s + 3 is s-hamiltonian if and only if κ(L(G)) ≥ s + 2. (ii)LetN2={Ns1,s2,s3 :s1\u3e0,s1≥s2≥s3≥0ands1+s2+s3≤4}.ThenforanyN∈N2, every N -free line graph L(G) with |V (L(G))| ≥ s + 3 is s-Hamilton-connected if and only if κ(L(G)) ≥ s + 3. 2. Results in matroids. A matroid M with a distinguished element e0 ∈ E(M) is a rooted matroid with e0 being the root. We present a characterization of all connected binary rooted matroids whose root lies in at most three circuits, and a characterization of all connected binary rooted matroids whose root lies in all but at most three circuits. While there exist infinitely many such matroids, the number of serial reductions of such matroids is finite. In particular, we find two finite families of binary matroids M1 and M2 and prove the following. (i) For some e0 ∈ E(M), M has at most three circuits containing e0 if and only if the serial reduction of M is isomorphic to a member in M1. (ii) If for some e0 ∈ E(M), M has at most three circuits not containing e0 if and only if the serial reduction of M is isomorphic to a member in M2. These characterizations will be applied to show that every connected binary matroid M with at least four circuits has a 1-hamiltonian circuit graph
The Computational Complexity of Some Games and Puzzles With Theoretical Applications
The subject of this thesis is the algorithmic properties of one- and two-player
games people enjoy playing, such as Sudoku or Chess. Questions asked about puzzles
and games in this context are of the following type: can we design efficient computer
programs that play optimally given any opponent (for a two-player game), or solve
any instance of the puzzle in question?
We examine four games and puzzles and show algorithmic as well as intractability
results. First, we study the wolf-goat-cabbage puzzle, where a man wants to transport
a wolf, a goat, and a cabbage across a river by using a boat that can carry only one
item at a time, making sure that no incompatible items are left alone together. We
study generalizations of this puzzle, showing a close connection with the Vertex
Cover problem that implies NP-hardness as well as inapproximability results.
Second, we study the SET game, a card game where the objective is to form
sets of cards that match in a certain sense using cards from a special deck. We
study single- and multi-round variations of this game and establish interesting con-
nections with other classical computational problems, such as Perfect Multi-
Dimensional Matching, Set Packing, Independent Edge Dominating Set,
and Arc Kayles. We prove algorithmic and hardness results in the classical and
the parameterized sense.
Third, we study the UNO game, a game of colored numbered cards where players
take turns discarding cards that match either in color or in number. We extend results
by Demaine et. al. (2010 and 2014) that connected one- and two-player generaliza-
tions of the game to Edge Hamiltonian Path and Generalized Geography,
proving that a solitaire version parameterized by the number of colors is fixed param-
eter tractable and that a k-player generalization for k greater or equal to 3 is PSPACE-hard.
Finally, we study the Scrabble game, a word game where players are trying to
form words in a crossword fashion by placing letter tiles on a grid board. We prove
that a generalized version of Scrabble is PSPACE-hard, answering a question posed
by Demaine and Hearn in 2008
Hamiltonian index is NP-complete
In this paper we show that the problem to decide whether the hamiltonian index of a given graph is less than or equal to a given constant is NP-complete (although this was conjectured to be polynomial). Consequently, the corresponding problem to determine the hamiltonian index of a given graph is NP-hard. Finally, we show that some known upper and lower bounds on the hamiltonian index can be computed in polynomial time
Hamiltonian index is NP-complete
In this paper we show that the problem to decide whether the hamiltonian index of a given graph is less than or equal to a given constant is NP-complete (although this was conjectured to be polynomial). Consequently, the corresponding problem to determine the hamiltonian index of a given graph is NP-hard. Finally, we show that some known upper and lower bounds on the hamiltonian index can be computed in polynomial time