Degree Deviation and Spectral Radius

Approximability of the independent feedback vertex set problem for bipartite graphs

CHAPTER 2 - Undirected Graphs

The graph coloring problem is one of the most famous problems in graph theory and has a large range of applications. It consists in coloring the vertices of an undirected graph with a given number of colors such that two adjacent vertices get different colors. This thesis deals with some variations of this basic coloring problem which are related to scheduling and discrete tomography. These problems may also be considered as partitioning problems. In Chapter 1 basic definitions of computational complexity and graph theory are presented. An introduction to graph coloring and discrete tomography is given. In the next chapter we discuss two coloring problems in mixed graphs (i.e., graphs having edges and arcs) arising from scheduling. In the first one (strong mixed graph coloring problem) we have to cope with disjunctive constraints (some pairs of jobs cannot be processed simultaneously) as well as with precedence constraints (some pairs of jobs must be executed in a given order). It is known that this problem is NP-complete in mixed bipartite graphs. In this thesis we strengthen this result by proving that for k = 3 colors the strong mixed graph coloring problem is NP-complete even if the mixed graph is planar bipartite with maximum degree 4 and each vertex incident to at least one arc has maximum degree 2 or if the mixed graph is bipartite and has maximum degree 3. Furthermore we show that the problem is polynomially solvable in partial p-trees, for fixed p, as well as in general graphs with k = 2 colors. We also give upper bounds on the strong mixed chromatic number or even its exact value for some classes of graphs. In the second problem (weak mixed graph coloring problem), we allow jobs linked by precedence constraints to be executed at the same time. We show that for k = 3 colors this problem is NP-complete in mixed planar bipartite graphs of maximum degree 4 as well as in mixed bipartite graphs of maximum degree 3. Again, for partial p-trees, p fixed, and for general graphs with k = 2 colors, we prove that the weak mixed graph coloring problem is polynomially solvable. We consider in Chapter 3 the problem of characterizing in an undirected graph G = (V, E) a minimum set R of edges for which maximum matchings M can be found with specific values of p = |M ∩ R|. We obtain partial results for some classes of graphs and show in particular that for odd cacti with triangles only and for forests one can determine in polynomial time whether there exists a minimum set R for which there are maximum matchings M such that p= |R ∩ M|, for p= 0,1, ..., ν(G). The remaining chapters deal with some coloring (or partitioning) problems related to the basic image reconstruction problem in discrete tomography. In Chapter 4 we consider a generalization of the vertex coloring problem associated with the basic image reconstruction problem. We are given an undirected graph and a family of chains covering its vertices. For each chain the number of occurrences of each color is given. We then want to find a coloring respecting these occurrences. We are interested in both, arbitrary and proper colorings and give complexity results. In particular we show that for arbitrary colorings the problem is NP-complete with two colors even if the graph is a tree of maximum degree 3. We also consider the edge coloring version of both problems. Again we present some complexity results. We consider in Chapter 5 some generalized neighborhoods instead of chains. For each vertex x we are given the number of occurrences of each color in its open neighborhood Nd(x) (resp. closed neighborhood Nd+(x)), representing the set of vertices which are at distance d from x (resp. at distance at most d from x). We are interested in arbitrary colorings as well as proper colorings. We present some complexity results and we show in particular that for d = 1 the problems are polynomially solvable in trees using a dynamic programming approach. For the open neighborhood and d = 2 we obtain a polynomial time algorithm for quatrees (i.e. trees where all internal vertices have degree at least 4). We also examine the bounded version of these problems, i.e., instead of the exact number of occurrences of each color we are given upper bounds on these occurrences. In particular we show that the problem for proper colorings is NP-complete in bipartite graphs of maximum degree 3 with four colors and each color appearing at most once in the neighborhood N(x) of each vertex x. This result implies that the L(1,1)-labelling problem is NP-complete in this class of graphs for four colors. Finally in Chapter 6 we consider the edge partitioning version of the basic image reconstruction problem, i.e., we have to partition the edge set of a complete bipartite graph into k subsets such that for each vertex there must be a given number of edges of each set of the partition incident to this vertex. For k = 3 the complexity status is still open. Here we present a new solvable case for k = 3. Then we examine some variations where the union of two subsets E1, E2 has to satisfy some additional constraints as for example it must form a tree or a collection of disjoint chains. In both cases we give necessary and sufficient conditions for a solution to exist. We also consider the case where we have a complete graph instead of a complete bipartite graph. We show that the edge partitioning problem in a complete graph is at least as difficult as in a complete bipartite graph. We also give necessary and sufficient conditions for a solution to exist if E1 ∪ E2 form a tree or if they form a Hamiltonian cycle in the case of a complete graph. Finally we examine for both, complete and complete bipartite graphs, the case where each one of the sets E1 and E2 is structured (two disjoint Hamiltonian chains, two edge disjoint cycles) and present necessary and sufficient conditions.

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial [Formula: see text], and any directed finite bipartite graph can be considered as a polynomial [Formula: see text], and vise verse. We also show that the multiplication in the semirings [Formula: see text], [Formula: see text] corresponds to an operation of the corresponding graphs. This operation is exactly the product of Petri nets in the sense of Winskel [G. Winskel and M. Nielsen, Models of concurrency, in Handbook of Logic in Computer Science, Vol. 4, eds. Abamsky, Gabbay and Maibaum (Oxford University Press, 1995), pp. 1–148]. As an application, we give an approach to dividing in the semirings [Formula: see text], [Formula: see text], and a criteria for parallalization of Petri nets. Finally, we endow the set of all bipartite graphs with the Zariski topology.

On the Carathéodory Number for the Convexity of Paths of Order Three

Let $G$ be a finite, simple, and undirected graph and let $S$ be a set of vertices of $G$. If no vertex of $G$ that does not belong to $S$ has two neighbors in $S$, then $S$ is $P_3$-convex. The $P_3$-convex hull $H_G(S)$ of $S$ is the smallest $P_3$-convex set containing $S$. The $P_3$-Caratheodory number of $G$ is the smallest integer $c$ such that for every set $S$ and every vertex $u$ in $H_G(S)$, there is a set $F\subseteq S$ with $|F|\leq c$ and $u\in H_G(F)$. We study structural and algorithmic aspects of the $P_3$-Caratheodory number. We characterize the $P_3$-Caratheodory number of trees and block graphs, establish upper bounds on the $P_3$-Caratheodory number of general graphs and of claw-free graphs, and prove that it is NP-complete to decide for a given bipartite graph $G$ and a given integer $k$ whether the $P_3$-Caratheodory number of $G$ is at least $k$.

A new kind of bipartite graph is defined in this paper. The work is motivated mainly from a graph introduced by Al-Kaseasbeh and Erfanian [A bipartite graph associated to elements and cosets of subgroups of a finite group, AIMS Math. 6(10) (2021) 10395–10404] and partially from the undirected power graph of a group and from the inclusion graph. For a nontrivial group [Formula: see text], we define a simple undirected bipartite graph [Formula: see text] with the vertex set [Formula: see text], where [Formula: see text] is the set of all elements of the group [Formula: see text] and [Formula: see text] is the set of all subgroups [Formula: see text] of [Formula: see text] such that [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here, we classify all the finite groups [Formula: see text] whose bipartite graphs [Formula: see text] are planar. In addition, we also classify the finite groups [Formula: see text] such that [Formula: see text] is outerplanar, maximal planar, maximal outerplanar, star, tree, claw graph, respectively. The planarity of the graph [Formula: see text] corresponding to an infinite group [Formula: see text] has also been investigated here. It is also observed that [Formula: see text] satisfies Vizing’s conjecture.

A sharpened condition for strict log-convexity of the spectral radius via the bipartite graph

This paper deals with embedded index coding problem (EICP), introduced by A. Porter and M. Wootters, which is a decentralized communication problem among users with side information. An alternate definition of the parameter minrank of an EICP, which has reduced computational complexity compared to the existing definition, is presented. A graphical representation for an EICP is given using directed bipartite graphs, called bipartite problem graph, and the side information alone is represented using an undirected bipartite graph called the side information bipartite graph. Inspired by the well-studied single unicast index coding problem, graphical structures, similar to cycles and cliques, are identified in the side information bipartite graph of a single unicast embedded index coding problem (SUEICP). Transmission schemes based on these graphical structures, called tree cover scheme and bi-clique cover scheme are also presented. For a class of SUEICPs, scalar linear optimal solution is given using bi-clique cover. A relation between connectedness of the side information bipartite graph and the number of transmissions required in a scalar linear solution of an EICP is established.

Let be a finite, simple and undirected graph. For any integer , generalized eccentricity power sum adjacency matrix of is matrix with its entry as and zero otherwise, where is the eccentricity of the vertex v of a graph G. In this paper, the new energy of graph under the name as generalized eccentricity power sum adjacency energy of has been introduced. Generalized eccentricity power sum adjacency energy of some standard graphs and regular graphs obtained by complete graph.
 Let be a finite, simple and undirected graph. For any integer , generalized eccentricity power sum adjacency matrix of is matrix with its entry as and zero otherwise, where is the eccentricity of the vertex v of a graph G. In this paper, the new energy of graph under the name as generalized eccentricity power sum adjacency energy of has been introduced. Generalized eccentricity power sum adjacency energy of some standard graphs and regular graphs obtained by complete graph.

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

The clique polynomial <span class="math"><em>C</em>(<em>G</em>, <em>x</em>)</span> of a finite, simple and undirected graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is defined as the ordinary generating function of the number of complete subgraphs of <span class="math"><em>G</em></span>. A real root of <span class="math"><em>C</em>(<em>G</em>, <em>x</em>)</span> is called a clique root of the graph <span class="math"><em>G</em></span>. Hajiabolhasan and Mehrabadi showed that every simple graph <span class="math"><em>G</em></span> has at least a clique root in the interval <span class="math">[ − 1, 0)</span>. Moreover, they showed that the class of triangle-free graphs has only clique roots. In this paper, we extend their result by showing that the class of <span class="math"><em>K</em>₄</span>-free chordal graphs has also only clique roots. In particular, we show that this class has always a clique root <span class="math"> − 1</span>. We conclude our paper with some interesting open questions and conjectures.

Let G(V, E) be a simple, connected and undirected graph. A dominating set $$S \subseteq V$$ is called a 2-secure dominating set (2-SDS) in G, if for each pair of distinct vertices $$v_1,v_2 \in V$$ there exists a pair of distinct vertices $$u_1,u_2 \in S$$ such that $$u_1 \in N[v_1]$$ , $$u_2 \in N[v_2]$$ and $$(S {\setminus } \{u_1,u_2\}) \cup \{v_1,v_2 \}$$ is a dominating set in G. The size of a minimum 2-SDS in G is said to be 2-secure domination number denoted by $$\gamma _{2s}(G)$$ . The 2-SDM problem is to check if an input graph G has a 2-SDS S, with $$ \vert S \vert \le k$$ , where $$ k \in \mathbb {Z}^+ $$ . It is proved that for bipartite graphs 2-SDM is NP-complete. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We reinforce the existing NP-complete result for bipartite graphs, by proving 2-SDM is NP-complete for some subclasses of bipartite graphs specifically, comb convex bipartite and star convex bipartite graphs. We prove that this problem is linear time solvable for bounded tree-width graphs. We also show that the 2-SDM is W[2]-hard even for split graphs. The M2SDS problem is to find a 2-SDS of minimum size in the given graph. We give a $$ \varDelta +1 $$ -approximation algorithm for M2SDS, where $$ \varDelta $$ is the maximum degree of the given graph and prove that M2SDS cannot be approximated within $$ (1 - \epsilon ) \ln (\vert V \vert ) $$ for any $$ \epsilon > 0 $$ unless $$ NP \subseteq DTIME(\vert V \vert ^{ O(\log \log \vert V \vert )}) $$ . Finally, we prove that the M2SDS is APX-complete for graphs with $$\varDelta =4.$$

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

Solving the weighted efficient edge domination problem on bipartite permutation graphs